fˇµ2012 c 2 8 fmath.sjtu.edu.cn/course/odehomepage/jc.pdf · ý›:3t...
TRANSCRIPT
~©§á(êÆ9'Ʀ^)
~©§ùÂ
Ü ?Í
þ°ÏÆ
#Fϵ2012c 2 8F
c ÷
~©§ß 17 V"ÄkÑy3ÃX Isaac Newton, Gottfried Leibniz
Ú Bernoulli[xÆ[ó¥. ¨ã¢S^l)
ÐÒNyÑ5,XÚî|^åÆnïá~©§,¿$^È©
y²/¥7$1;´ý.
é¢S¯K (AO´ÚîåÆ!UNåÆ)¥Ñþ©
§¦)I,r?ÚíÄÈ©)ÚuÐ.Ï ~©§
´éXÈ©;§,§´éõêÆY§Ä:.
XEuÐ,~©§3éõ'Æ,XÔn!ó§!>f&E!
)ÔÚzƱ9²L7K,¥u^©
~©§3éuÐÏSѴϦ«). 1841 c Jeseph
Liouvilley²a/ªþ~ü Riccati§ØU^ÐÈ©¦
). Liouvilleór¦<Ϧ#ïÄ~©§ÚnØ. 19
V" 20VÐ, Henri PoincareXmM5óC½yÄåX
ÚÄ:.
Ö´?öõc53þ°ÏÆêÆX),±9óÚ²L
+nX)ÇùÂÄ:þ?. ùÂÀáK´¦þ
ò~©§Ä:7nØNXÀ\á, Ó3Ù!¥
B0ÄåXÚCnØÐÚ£. 3Ä:nØy²þQ
À^²;©ÛÔöÆ)©ÛÚíUå, qÚ\Ü©
y©ÛÔöÆ)ÄgÚÜ6ínUå, ¿¦¦éy
©ÛkÐÚ@£.
ÖQãÚy²åfw´Ã, BÆ)gÆ. Ö¥kþ
5P,ùÑ´~SN.§éSNn)äkéÐ
Ï.F"Öö3ÖÖL§¥Àé5PÜ©n).
ÐÈ©Ü©0Aa§¦), ¿kþ~K.
ý:3T§ÚÈ©Ïf±95©§, ÏùÜ©SN
Ø=3þ 3nØþÑ´é. ٧ܩѴÑù,Ï
éõ§Ñ±ÏL MathematicaÚ MapleêÆ^5¦). ÏdÖ¥
ÏL~fü/0 Mathematica^.
~©§Ä:nØÌ©Ù31ÙÚ1nÙ. 1Ù:ã
Xþ~©§Ð¯K)35!5,±9)'ugCþ!Ð
©^ÚëêëY65. ùÜ©SNy²Ìóä´ål
m¥Ø NnÚ Arzela-AscoliÚn.
1nÙÌ9p©§Ú§|)Ä:nØ.äN)
35,)'ugCþ!Щ^ÚëêëY5;)Û
©§|ÛÜ)Û)35; ©§|ÛÜÈnر9§3
5Ú[5 ©§¦)¥A^.
1oÙÌùã5©§|Úp5©§)3«
mÚÏ)(, ~Xê5©§|Úp~Xê5©§
),±9CXê5©§|Ä:nØ.¿|^~Xê
5©§|)Ú Mathematicsã?ز¡àg5©§
")ÛÜÿÀ(. CXꩧܩý:´±ÏXê5
©§|Ä)Ý FloquetIO., 9±Ï) Floquet¦êm'
X;5©§)":!>¯KÚ?ê). AO/,
Ùép~Xê5©§Ä)|Ñ'y². é)
Û5©§)Âñ»Ñ#y².
1ÊÙ´~©§nØCÜ©. 0k')½
5,43ÚØ3ü½,±9©|¯KeZVgÚ~
f.
á±ÆÏùÇ.éØÓÆéÚÆê,1nÙ
©§)Û)ÚÈnØ, 1oÙCXê5©§Ä:nØ
Ú1ÊÙ²¡g£©XÚSN±ÀJÜ©ÆS, ÏùÙ
!SNéÕá,ØKÙ§Ù!SNÆS.
Ù´N¹,Ù¥Ñ Arzela-AscoliÚny²ÚÛÉ¢Ý
2
Ýéê35y². Ï Arzela-Ascoli Ún9Ùy²Ñ3
¼©ÛÖÄm¥Ñ,$c?Æ)Öå5(J.
BÖö,N¹¥Ñüy². ¢ÛÉÝÝéê
35õê5êÖvkÑy², BÖö3N¹
¥ÑÙy².
Öë©zØ=~©§'Ö7Ú©z,
ÄåXÚ©zÚÖ7. 8´4ÆkåÆ)ÏL?Ú
'©z],é~©§ynØkÐÚ).
ùÂ3þ°ÏÆêÆXÚ²L+nÆ)Á^L§
¥, éõÓÆuyØ<Ø, ¿JÑB?U¿ÚïÆ.
3d¦L«©%a.
duY²Úm, Ö¥UkaØv$Ø/.
ÖöeuyÖ¥?Û¯KÑU9?ö6, ±B?Ú?
U.Øa-
?Íö
2012c 2 8F
3
8 ¹
1Ù ~©§Ä:£ 1
1 ~©§ÄVg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 ©§Ú) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ©§Ú)~f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 ©§)AÛ)º!3Ú5 . . . . . . . . . . . . . . . . . . . 4
1.4 ¢S¯K.í . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 ÐÈ© . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 T§ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 È©Ïf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Aa=zT§§ . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 5©§~êC´ . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Ûª§ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 p©§ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Mathematica ¦)~©§ . . . . . . . . . . . . . . . . . . . . . . . . 30
3 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1Ù ©§)35Ú5 37
1 ý£µålmØ Nn . . . . . . . . . . . . . . . . . . . . . . . . 37
1.1 ålm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.2 Ø Nn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 )35: Picard ½n . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 )35µPeano ½n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 )éÐÚëêëY65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1nÙ p©§Ú§|)nØ 54
1 p©§Ú§|: )5 . . . . . . . . . . . . . . . . . . . . . . . . 54
2 )Û©§)Û) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4
~©§
3 ©§ÈnØ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1 ÈÄ:nصÄgÈ©35 . . . . . . . . . . . . . . . . . . . . 65
3.2 ÄgÈ©3 ©§¦)¥A^ . . . . . . . . . . . . . . . . . . . . 70
3.3 Hamilton XÚÈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 SKn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
1oÙ 5©§ÄnØÚ) 83
1 5©§)ÄnØ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1.1 5©§|)3«m . . . . . . . . . . . . . . . . . . . . . . . . 84
1.2 5©§|Ï)( . . . . . . . . . . . . . . . . . . . . . . . . . 85
1.3 p5©§Ï)( . . . . . . . . . . . . . . . . . . . . . . . . 90
2 ~Xê5©§|) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.1 Ýê¼ê~Xê5©§|) . . . . . . . . . . . . . . . . 96
2.2 ~Xêàg5©§|Ä)ݦ . . . . . . . . . . . . . . . . 98
2.3 A^µ²¡~Xê5©XÚÛÜ( . . . . . . . . . . . . . . . . 105
2.4 ^ Mathematica ¦§|)Ú²¡©§)ÛÜã . . . . . . 110
3 p~Xê5©§) . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.1 ~Xêàg5©§) . . . . . . . . . . . . . . . . . . . . . . . 111
3.2 ~Xêàg5©§½Xê . . . . . . . . . . . . . . . . . 116
4 CXê5©§Ä:nØ . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.1 ±ÏXê5©§|µFloquet nØ . . . . . . . . . . . . . . . . . . 118
4.2 CXêàg5©§: '½nÚ Sturm-Liouville >¯K . 124
4.3 pCXê5©§µ?ê) . . . . . . . . . . . . . . . . . . . 133
5 SKo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
1ÊÙ ©§½5Ú½5nØ 147
1 ©§)½5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
1.1 àg5©§|")½5 . . . . . . . . . . . . . . . . . . . . . 148
1.2 d5Cq(½5§½5 . . . . . . . . . . . . . . . . . . . 152
1.3 ½½5 Lyapunov 1 . . . . . . . . . . . . . . . . . . . . . . 153
5
8¹
2 ²¡g£©XÚ: 4Ú©| . . . . . . . . . . . . . . . . . . . . . . . . . . 157
2.1 435Ú½5½ . . . . . . . . . . . . . . . . . . . . . . . . 159
2.2 ©|¯KAü~f . . . . . . . . . . . . . . . . . . . . . . . . . 164
3 SKÊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
18Ù N¹ 171
1 Arzela-Ascoli Úny² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
2 Ýéê35y² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
ë©z 175
6
Î Ò L
R ¢ê8£½¢ê¤
C Eê8£½Eê¤
Z ê8£½ê+¤
Rn ¢n5m
A \B áu8ÜAØáu8ÜB¤8Ü
∃ 3
∀ ?¿
n! 1 n ë¦È, = n(n− 1) · 2 · 1
x ∈ D x áu8Ü D
f ∈ C(Ω) f ´« Ω þëY¼ê
f ∈ Ck(Ω) f ´« Ω þ k gëY¼ê
⇒ Âñ
A ⊂ B 8ÜA´8ÜBf8
C[a, b] ½Â3 [a, b]þëY¼êN¤8Ü
A⇐⇒ B ·KAdu·KB
D(f1, . . . , fn)
D(y1, . . . , yn)¼ê fi(x, y1, . . . , yn), i = 1, . . . , n 'u y1, . . . , yn Jacobi 1ª
∂(f1, . . . , fn)
∂(y1, . . . , yn)¼ê fi(x, y1, . . . , yn), i = 1, . . . , n 'u y1, . . . , yn Jacobi Ý
(Rn,0) Rn m¥I: 0 ,
fx(x, y) ¼ê f(x, y) 'uÙCþx ê
7
1Ù ~©§Ä:£
ÙÌ0~©§Ä£, X©§Ú)½Â, )AÛn),
)3ÚQã, ±9©§«). ©§nØÜ©ÙØ9.
§1.1 ~©§ÄVg
!Äk0©§, ±9)ÚÏ)½Â
§1.1.1 ©§Ú)
©§´¹k¼êê§. ¼êgCþ´üCþ©§¡
~©§. ¼êgCþ´õCþ©§¡ ©§. ©§¹k
êpê¡©§.
~K:
1. §d3y
dx3+ (y5 + xy + 1)
dy
dx= 1 ´ 3 ~©§.
2. § x2 d4x
dt4+
(dx
dt
)5
= cosx ´ 4 ~©§.
3. Newton 1$ĽÆÑ©§ md2x(t)
dt2= F (x(t) ´ 2 ~©§, Ù¥ m
´:þ, F ´ t âf3 x(t) É^å.
4. §∂2u(x, y)
∂x2+∂2u(x, y)
∂y2= 0 ´ 2 ©§.
5. §∂u(x, y, z)
∂x+ (u+ 1)
∂u(x, y, z)
∂y− xyz ∂u(x, y, z)
∂y= u3 ´ 1 ©§.
ÖÌùã~©§, ~©§nØ3 ©§¥A^, Ö¥ §3.3.2 0
©§).
n ~©§/ª´
F
(t, x(t),
dx
dt(t), . . . ,
dnx
dtn(t)
)= 0, (1.1.1)
1
§1.1 ~©§ÄVg
Ù¥ F ´'u n+ 2 Cþ½¼ê, F 7L¹kdnxdtn . Ï x 'u t n ê¹3
¼ê F ¥, ¤±¡ (1.1.1) n Ûª~©§ (¡ n Ûª§). ±B
å~^ x, x, x′(t), x′′(t) Ú x(n)(t) L«¼ê x 'ugCþ t ê. ~©
§¥, S.þ~^m t gCþ; ~^ y ÏCþ, x gCþ.
n wª~©§/ª´
x(n)(t) = f(t, x(t), x′(t), . . . , x(n−1)(t)
), (1.1.2)
Ù¥ f ´'u n + 1 Cþ¼ê. ´, w«~©§±¤Ûª~©§/
ª. Ûª~©§ÛÜ/±|^Û¼ê3½n¤w«/ª.
¼ê F ½Â3 Rn+2 m,m« Ω þ. ½Â3 (t1, t2) þ¼ê x = φ(t) ¡
©§ (1.1.1) ), XJ φ(t) 3 (t1, t2) þäk n− 1 ëYê, Ù n ê3, (t, φ(t), φ′(t), . . . , φ(n)(t)
)∈ Ω,
F(t, φ(t), φ′(t), . . . , φ(n)(t)
)≡ 0, t ∈ (t1, t2).
¡ (t1, t2) )½Â«m. 5µkU t1 = −∞ ½ t2 =∞.
Λ ⊂ Rn ´m«, c = (c1, . . . , cn) ∈ Λ. ¹k n ~ê¼ê x = φ(t, c), (t, c) ∈
(t1, t2)×Λ, ¡§ (1.1.1)Ï),XJ φ´§ (1.1.1)), n~ê´?¿½Õ
á, = φ, φ′, . . . , φ(n−1) 'u c1, c2, . . . , cn Jacobi 1ª
D(φ, φ′, . . . , φ(n−1))
D(c1, c2, . . . , cn):=
∣∣∣∣∣∣∣∣∣∣∣∣∣
∂φ∂c1
∂φ∂c2
· · · ∂φ∂cn
∂φ′
∂c1
∂φ′
∂c2· · · ∂φ′
∂cn...
.... . .
...
∂φ(n−1)
∂c1
∂φ(n−1)
∂c2· · · ∂φ(n−1)
∂cn
∣∣∣∣∣∣∣∣∣∣∣∣∣6= 0, (t, c) ∈ (t1, t2)× Λ.
n ©§ (1.1.1) ½ (1.1.2) ÷vЩ^
x(t0) = x0, x′(t0) = x1, . . . , x
(n−1) = xn−1, (1.1.3)
¡Ð¯K, Ù¥ t0 ∈ R ¡Ð©m, (x0, x1, . . . , xn−1) ∈ Rn ¡Ð©½¡Ð.
©§ (1.1.1) ½ (1.1.2) ÷vЩ^ (1.1.3) )¡Ð¯K).
N5:
• n ©§Ð¯K¥Ð©^´d n ^(½.
2
1Ù ~©§Ä:£
• Щ (x0, x1, . . . , xn−1)áuþ¼ê (φ(t, c), φ′(t, c), . . . , φ(n−1)(t, c)), (t, c) ∈ t0×
Λ, , ЯK)¹3Ï)¥. ù´Ï|^Û¼ê3½n±)Ñ
A c0,l φ(t, c0)ҴЯK). ~X y = cex ´©§ y′ = y 3 RþÏ
). §¹§¤k).
• e~fL²Ï)7¹©§¤k). ~X©§
dy
dx= y2,
kÏ) y = −(x + c)−1, Ù¥ c ´?¿~ê. w, y = 0 ´§), §Ø¹3
Ï)¥.
§1.1.2 ©§Ú)~f
1. ©§
x′′(t) = g, g ∈ R,
3 t ∈ R þkÏ) x = φ(t, c1, c2) = 12gt
2 + c1t+ c2, Ù¥ c1 Ú c2 ´?¿~ê.
2. n©§
x′′′(t) + x′′(t)− x′(t) + 15x(t) = 0,
3 t ∈ RþkÏ) x = φ(t, c1, c2, c3) = c1e−3t + c2e
t cos(2t) + c3et sin(2t), Ù¥ c1, c2, c3
´?¿~ê.
3. a(x), b(x) 3 (α, β) ⊂ R þëY, x0 ∈ (α, β), y0 ∈ R. K©§Ð¯K
dy
dx= a(x)y + b(x), y(x0) = y0,
3 x ∈ (α, β) þk)
y(x) = e∫ xx0a(s)ds
(y0 +
∫ x
x0
b(t)e−
∫ tx0a(s)ds
dt
).
4. ЯK
dy
dx= y
13 , y(1) = 0,
3
§1.1 ~©§ÄVg
3 x ∈ R þkáõ)
y(x) =
0, x ≤ c,
±(
23
) 32 (x− c) 3
2 , x > c,
Ù¥ c ≥ 1 ´?¿~ê.
5. ©§
dy
dx= y2,
– ÷vЩ^ y(1) = 1 3 (−∞, 2) þk) y = (2− x)−1;
– ÷vЩ^ y(1) = −1 3 (0,∞) þk) y = −x−1.
6. ЯK
dy
dx= 1 + y2, y(0) = 0,
3 (−π2 ,π2 ) þk) y = tanx.
N5:
• ~ 4 ¥§mà¼ê3 (x, y) ²¡ëY, 3 y = 0 Ø, ЯKkáõ).
ù9©§Ð¯K)35¯K, ò3©¥ÅÚ0.
• ~ 5 Ú 6 ¥©§mà¼ê3 (x, y) ²¡þëY, )½Â«mké«
O. ù´©ò0)òÿ¯K.
§1.1.3 ©§)AÛ)º!3Ú5
ãÄnØ, ÄkÄXew«©§
x = f(t, x), (1.1.4)
Ù¥ f 3 R2 ,m« Ω þëY. x = φ(t), t ∈ (α, β) ´§ (1.1.4) ). K
(t, φ(t)) : t ∈ (α, β) ´ Ω ¥^1w(Ùy²ò3eÙÑ), ¡§ (1.1.4)
È©. 3È©þ?: (t0, φ(t0)), ÙÇ φ′(t0) u f(t0, φ(t0)). ù`²é
u Ω ¥?: (t, x), XJkÈ©ÏL, KÏLT:È©Ç f(t, x).
4
1Ù ~©§Ä:£
é ∀ (t, x) ∈ Ω, LT:Ç f(t, x) ã. Ω ¥¤kùãN¤
8Ü¡§ (1.1.4) |. |^|±Cq/ѧȩ. |
*~f´^/^c±^|µ3[äkKü4^/^c±gþá
c,§òUì^|3^c±ü. ¤kùk5KücÒ¤
^|¤÷v~©§|. ÖØäNïá^/^c^|÷v§,k,
Ööë [16, p.16, ~3].
¯K: ½ (t0, x0) ∈ Ω, § (1.1.4) ÷vЩ^ x(t0) = x0 )´Ä3ºXJЯ
K)3, @o)´Äº
5: ±Bå, ò§ (1.1.4) ÷vЩ^ x(t0) = x0 )`¤§ (1.1.4) LÐ
©: (t0, x0) ), ½§ (1.1.4) LЩ: (t0, x0) È©.
þã¯K)û3~©§uФþ²ém. IêÆ[Augustin Cauchy
(1789–1857) u 19 V 20 cïá~©§Ð¯K)35½n (ÏX
d,ЯKq¡Cauchy¯K).IêÆ[ Rudolf Lipschitz (1832–1903)u 1876c~f
Cauchy'uЯK)35½n^.5IêÆ[ Charles Emile Picard
(1856–1941) Ú ¥=êÆ[ Ernst Lindelof (1870–1946) Ñ Lipschitz (J#y², AO
´ Picard u 1893 c^Åg%Cy² Lipschitz ½n (5õêÖÑ^ Picard
y²,ÏdT½nq¡ Picard ½n,½ Cauchy–Lipschitz ½n,½ Picard–Lindelof
½n). Peano ° Picard ½n^, y²ëY5=y)35 (,=këY
5Ãy5). Peano(J<¡ Peano ½n. 'u~©§)3Ú
5kéõÙ§?Úí2ÚU?, ÑÖ£, Ø3d0.
e¡½ny§ (1.1.4) ЯK)35Ú5.
½n1. (Picard½n) f(t, x) 3m« Ω ⊂ R2 þëY, 'u x ÷vÛÜ Lipschitz ^
, =é ∀ (t, x) ∈ Ω, 3 (t, x) Ut,x, 9~ê Lt,x, ¦é ∀ (t, x1), (t, x2) ∈ Ut,x Ñ
k
|f(t, x1)− f(t, x2)| ≤ Lt,x|x1 − x2|.
K§ (1.1.4) L?: (t0, x0) ∈ Ω Ñk), P x = φ(t), t ∈ (α, β).
N5:
• þã½nÑ©§ (1.1.4) ЯK)3Ú5¿©^. XJòÛÜ
Lipschitz ^¤ f(t, x) 'u x 3 Ω ¥këY ê, K½n(Øw,¤á.
5
§1.1 ~©§ÄVg
• T½nyl Ω ¥?:Ñuk=k^È©ÏL. ?Ú/,
– XJ x = ψ(t), t ∈ (µ, ν) ´§ (1.1.4) L (t0, x0) ), ÷v (α, β) ( (µ, ν),
t ∈ (α, β) ψ(t) = φ(t), ¡) x = ψ(t) ´) x = φ(t) òÿ.
– eéu§ (1.1.4)L (t0, x0)?) x = ψ(t), t ∈ (µ, ν)Ñk (µ, ν) ⊂ (α, β), ¡
) x = φ(t), t ∈ (α, β) ´§ (1.1.4) L (t0, x0)Øòÿ); d¡ (α, β) ´
§ (1.1.4) L (t0, x0))3«m½¡3«m.
• 3þ!~ 1, 2, 3, 4 ¥)3«mÑ´ (−∞,∞), ~ 5 ¥)3«m´k
, ~ 6 ¥)3«m´k. 5¿, ù)Ñ´Øòÿ).
½n 1 Ø=y)35, y)5. ¯¢þ, f(t, x) ëY5Ò
y)35.
½n2. (Peano ½n) XJ f(t, x) 3m« Ω ⊂ R2 þëY, K§ (1.1.4) L?:
(t0, x0) ∈ Ω )Ñ3, )½Â33«mþ.
½n 1 Ú 2 y²ò31Ù¥Ñ.
þ!~ 4 `²=këY5ØUyЯK)5. XJ f(t, x) Ø÷vëY5
b, ЯK)35ØUy. ~X, ЯK
dy
dx= f(x, y), y(0) = 0, (x, y) ∈ (R2,0). (1.1.5)
vk), Ù¥ (R2,0) L«I:,,
f(x, y) =
1, (x, y) ∈ (R2,0), (x, y) 6= (0, 0)
0, (x, y) = (0, 0).
¯¢þ, $^y, XJЯK (1.1.5) k), P y = φ(x), x ∈ J = [0, β) (3 0
>«m±aq/?Ø). Uì)½Â, φ(x) 3 J þëY, ê3. du3:,
dy
dx= 1, ¤±3: φ(x) = x + b. qÏ φ(x) ëY, ¤± 0 = φ(0) = lim
x→0φ(x) = b.
φ(x) = x. Ï φ′(0) = 1. ùdφ(x)
dx
∣∣∣∣x=0
= f(0, φ(0)) = f(0, 0) = 0 gñ. ùgñ`²Ð
¯K (1.1.5) vk).
6
1Ù ~©§Ä:£
§1.1.4 ¢S¯K.í
1. gdáN$Ä ±/¡I:, YþI¶ x . , t0
3pÝ x0 /òþ m ÔNe, XØÄÙ¦å3å^eÔN3 t
(> t0) XÛ? bå\Ý g.
ÄåÚ x , ¿$^Úî1½Æ mx = −mg, =
x(t) = −g.
ù´LãÔNgdeá$ħ.
2. 5ÔPC k,5Ô3 t êþ x(t), ÔPCÝÔ
êþ¤', Kk
x(t) = −kx(t),
Ù¥ k > 0 ´ÔPC'~~ê. T~£ã5ÔPC5Æ.
3. <. <¯K´~E,, §éX¬ÆÚ)ÔÆÃõÏ. ~Ä
nz¹.
<3, t êþ x(t), <OÇ k(t, x). K<C5Æ
x(t) = k(t, x)x(t).
5µ<OÇÑ)ÇÚk, §K.
3] 4Ù´L¹e, <OÇ k ±w~ê. ù´ Malthus <nØÄ
:. ¯¢þ, X<O5/«¸», ±9<mék] p¿, <
ODZ k = a(1− xL ), Ù¥~ê L¡¸Nþ. T.¢S/N<
Cz. ,éuäN¢S¹, <ODZ٧ܷ¼ê.
4. êÆ$Ä êÆ´nþ m l ü, Ù3å^e$Ä. 5½
lmYmÝ, x üYY. 3êÆ$Ä
$^Úî1½Æ
mlx = −mg sinx,
=
x = −ml
sinx.
T§£ãü$Ä5Æ.
7
§1.2 ÐÈ©
5. RLC £´>6 Äd> (R)!>a (L)!>N (C)Ú> ¤Gé>´. 3 t
> >Ø E(t), >´>6rÝ I(t). KT>!>aÚ>Nüà>Ø©O
´
UR(t) = RI(t), UL(t) = LI ′(t), UC(t) =1
C
∫I(s)ds.
d>´ó Kirchhoff ½Æ
UR(t) + UL(t) + UC(t) = E(t).
é§ü>'u t ¦ê'u£´>6÷v©§
LI ′′(t) +RI ′(t) +1
CI(t) = E′(t).
±þÑåÆ!>ÆÚ)ÔÆþü¢S.÷vÄåƧí. 3
e¡ÆL©§¦), ÏL¦)?Ún)Úݺ¢S¯K$Ä5Æ.
§1.2 ÐÈ©
!0ÏLÐÈ©Ò±¦)Aa~©§¦).
§1.2.1 T§
ò©§
dy
dx= f(x, y),
¥gCþ x ÚÏCþ y w¤é, KT§±¤é¡/ªµ
f(x, y)dx− dy = 0.
!Äé¡/ª©§
P (x, y)dx+Q(x, y)dy = 0, (1.2.1)
Ù¥ P (x, y) Ú Q(x, y) 3m« Ω ⊂ R2 ¥ëY.
§ (1.2.1) ¡T§½©§, XJ3 Ω þ¼ê Φ(x, y) ¦
dΦ(x, y) = P (x, y)dx+Q(x, y)dy, (x, y) ∈ Ω.
8
1Ù ~©§Ä:£
d¡ Φ(x, y) = c (c ?¿~ê) ´§ (1.2.1) ÏÈ©. 5µ(1.2.1) ´T§du
3¼ê Φ(x, y) ¦
Φx(x, y) = P (x, y), Φy(x, y) = Q(x, y), (x, y) ∈ Ω,
Ù¥ Φx Ú Φy ©OL« Φ 'u x Ú y ê.
e¡·KL²ÏÈ©3¦)¥^.
·K3. e Φ(x, y) = c (c ?¿~ê) ´§ (1.2.1) 3 Ω SÏÈ©, K Φ(x, y) = c )
=§´ (1.2.1) 3 Ω ¥).
y: 75. Ï Φx Ú Φy ØÓ", Ø Φy 6= 0. éu c ∈ R, y = u(x), x ∈ I ´
âÛ¼ê3½nl Φ(x, y) = c ¥¦Ñ). Kk Φ(x, u(x)) ≡ c, x ∈ I. l
0 ≡ dΦ(x, u(x))
dx= Φx(x, u(x)) + Φy(x, u(x))u′(x),
=
P (x, y)dx+Q(x, y)dy|y=u(x) = Φx(x, u(x))dx+ Φy(x, u(x))du ≡ 0, x ∈ I.
ù`²l Φ(x, y) = c ¥¦Ñ¼ê y = u(x) ´§ (1.2.1) ).
¿©5. y = u(x), x ∈ I ½ x = v(y), y ∈ J ´§ (1.2.1) 3 Ω S). ØÄc
ö, y Φ(x, u(x)), x ∈ I ðu~ê. ¯¢þ, Ï Φ(x, y) 3 Ω S, ¤±
dΦ
dx(x, u(x)) = Φx(x, u(x)) + Φy(x, u(x))u′(x) = P (x, u(x)) +Q(x, u(x))u′(x) ≡ 0, x ∈ I.
ùÒy²÷X§ (1.2.1) 3 Ω S?) Φ(x, y) Ñ~. ·Ky..
Q,ÏÈ©3é¡/ª©§¦)¥åXXd^, @o¯K´
• XÛ½§ (1.2.1) ´Ä´T§º
• XJ (1.2.1) ´T, XÛ¦ÏÈ©º
½n4. (T§½) P (x, y), Q(x, y) 9 ê Py(x, y), Qx(x, y) 3Ý/«
R ⊂ R2 þëY. K (1.2.1) ´T§=
Py(x, y) = Qx(x, y), (x, y) ∈ R. (1.2.2)
9
§1.2 ÐÈ©
y: 75. db, 3 R þ¼ê Φ(x, y) ¦
Φx(x, y) = P (x, y), Φy(x, y) = Q(x, y), (x, y) ∈ R.
k
Py(x, y) = Φxy(x, y), Qx(x, y) = Φyx(x, y), (x, y) ∈ R,
Ù¥ Φxy =∂2Φ(x, y)
∂x∂y, Φyx =
∂2Φ(x, y)
∂y∂x.
db Φxy(x, y) Ú Φyx(x, y) 3 R þëY, ¤± Φxy(x, y) = Φyx(x, y), (x, y) ∈ R. l
Py(x, y) = Qx(x, y), (x, y) ∈ R.
¿©5. ? (x0, y0) ∈ R. -
Φ(x, y) =
∫ x
x0
P (s, y)ds+
∫ y
y0
Q(x0, t)dt. (1.2.3)
K Φx = P (x, y). Ï Py(x, y) = Qx(x, y), ¤±
Φy(x, y) =
∫ x
x0
Py(s, y)ds+Q(x0, y) =
∫ x
x0
Qx(s, y)ds+Q(x0, y) = Q(x, y).
y..
N5:
• 3üëÏ« R þ, 3^ (1.2.2) e, È©
Φ(x, y) =
∫ (x,y)
(x0,y0)
Pdx+Qdy,
È©´»Ã'. Ïd3¦ Φ(x, y) , ±À¦O¦þüЩ: (x0, y0)
Úl (x0, y0) (x, y) ´uO´». 'Xkl (x0, y0) (x, y0), 2l (x, y0)
(x, y) ´»ÏÈ©
Φ(x, y) =
∫ x
x0
P (s, y0)ds+
∫ y
y0
Q(x, t)dt.
• ½n 4 ¥Ý/« R ±´?ÛüëÏà«.
~K: ½§
(yex + 2ex + y2)dx+ (ex + 2xy)dy = 0,
´Ä´T§. XJ´T§, ¦ÙÏÈ©.
10
1Ù ~©§Ä:£
): - P = yex + 2ex + y2, Q = ex + 2xy. Kk Py = Qx = ex + 2y. T§.
(0, 0) Щ:, À´» (0, 0) −→ (x, 0) Ú (x, 0) −→ (x, y). KkÏÈ©
Φ(x, y) =
∫ x
0
P (s, 0)ds+
∫ y
0
Q(x, t)dt = 2ex + exy + xy2 − 2.
N5: T§k±^©|Ün©¦). 'X
(exy + y2 cosx
)dx+ (ex + 2y sinx+ y) dy
= (exydx+ exdy) +(y2 cosxdx+ 2y sinxdy
)+ ydy
= d(exy) + d(y2 sinx) + d
(1
2y2
).
§1.2.2 È©Ïf
1. È©Ïf
é¡/ª§ (1.2.1)UØ´T§,k¦±,ðØ"Ïf µ(x, y)
¤T§, =
µ(x, y)P (x, y)dx+ µ(x, y)Q(x, y)dy = 0, (x, y) ∈ Ω, (1.2.4)
´T§. ¡ µ(x, y) ´§ (1.2.1) 3 Ω ¥È©Ïf. 5¿, § (1.2.1) Ú (1.2.4)
3 Ω þ´Ó) (Ï µ(x, y) 6= 0), ¦)§ (1.2.1) du¦)§ (1.2.4).
¯K: § (1.2.1) ´Ä3È©ÏfºXÛ½º
·K5. § (1.2.1) 3 Ω þkÈ©Ïf='u µ ©§
P∂µ
∂y−Q∂µ
∂x=
(∂Q
∂x− ∂P
∂y
)µ,
3 Ω þk"). AO/,
• § (1.2.1) k¹ x È©Ïf¿^´
Py(x, y)−Qx(x, y)
Q(x, y),
´ x ¼ê, P G(x), KÈ©Ïf µ(x) = e∫G(x)dx.
11
§1.2 ÐÈ©
• § (1.2.1) k¹ y È©Ïf¿^´
Qx(x, y)− Py(x, y)
P (x, y),
´ y ¼ê, P H(y), KÈ©Ïf µ(y) = e∫H(y)dy.
y: dT§½Â±í, lÑ. 5: ©¥")´Øðu").
~K: ¦)§
dy
dx+ p(x)y = q(x)yn, n ≥ 0, (1.2.5)
Ù¥ p(x), q(x) ´,m«m (α, β) þëY¼ê.
): Äkò§ (1.2.5) ¤é¡/ª
(p(x)y − q(x)yn)dx+ dy = 0. (1.2.6)
- P (x, y) = p(x)y − q(x)yn, Q(x, y) = 1.
XJ n = 0, § (1.2.5) ¡5§. Ï (Py −Qx)/Q = p(x), § (1.2.6) kÈ©Ï
f µ(x) = e∫p(x)dx. l ÏÈ©
Φ(x, y) = ye∫p(x)dx −
∫q(x)e
∫p(x)dxdx.
5§ (1.2.5) Ï)
y = e−∫p(x)dx
(c+
∫q(x)e
∫p(x)dxdx
), Ù¥ c ´?¿~ê.
XJ n = 1,§ (1.2.5)¡àg5§. Ï (Py−Qx)/Q = p(x)− q(x),§ (1.2.6)
kÈ©Ïf µ(x) = e∫
(p(x)−q(x))dx. l ÏÈ©
Φ(x, y) = ye∫
(p(x)−q(x))dx.
àg5§ (1.2.5) Ï)
y = ce−∫
(p(x)−q(x))dx, Ù¥ c ´?¿~ê.
XJ n > 0 n 6= 1, § (1.2.5) ¡ Bernoulli §. dØU$^·K 5. ´
y = 0 ´§ (1.2.5) ). y 6= 0 , 顧 (1.2.6) du
(p(x)y1−n − q(x))dx+ y−ndy = 0. (1.2.7)
12
1Ù ~©§Ä:£
- P (x, y) = p(x)y1−n − q(x), Q(x, y) = y−n. Kk (Py − Qx)/Q = (1 − n)p(x), ¤±§
(1.2.7) kÈ©Ïf µ(x) = e∫
(1−n)p(x)dx. é§
e∫
(1−n)p(x)dx(p(x)y1−n − q(x))dx+ y−ne∫
(1−n)p(x)dxdy = 0,
È©(1.2.7) ÏÈ©
Φ(x, y) = y1−ne∫
(1−n)p(x)dx − (1− n)
∫q(x)e
∫(1−n)p(x)dxdx.
¤± Bernoulli §k) y = 0 Ú ÏÈ©
y1−ne∫
(1−n)p(x)dx − (1− n)
∫q(x)e
∫(1−n)p(x)dxdx = c,
Ù¥ c ´?¿~ê.
N5:
• 5§ÏÈ©±l Bernoulli §ÏÈ©¥- n = 0 .
• Bernoulli § y 6= 0 ±ÏLC z = y1−n =z5§
dz
dx+ (1− n)p(x)z = (1− n)q(x).
2. ©|ܦȩÏf
½é¡/ª©§ (1.2.1), kØU^·K 5 ½È©Ïf
35, I©|ÜéÈ©Ïf. ~X, §
y3dx+ 2(x2 − xy2)dy = 0, (1.2.8)
éJ^®k¦). ò§ (1.2.8) ©|Ü
(y3dx− 2xy2dy) + 2x2dy = 0,
ØJuycökÈ©Ïf µ1(y) = y−5, ökÈ©Ïf µ2(x) = x−2. XÛ|^üöÈ©
Ïf (1.2.8) È©Ïf.
·K6. éuÈ©Ïf, e(ؤá.
13
§1.2 ÐÈ©
• µ(x, y) ´ (1.2.1) È©Ïf, k
µ(x, y)P (x, y)dx+ µ(x, y)Q(x, y)dy = dΦ(x, y),
Ké?¿ëY¼ê g(·), µ(x, y)g(Φ(x, y)) ´ (1.2.1) È©Ïf.
• ò§ (1.2.1) ©|¤
(P1(x, y)dx+Q1(x, y)dy) + (P2(x, y)dx+Q2(x, y)dy) = 0.
e3 µ1(x, y),Φ1(x, y), µ2(x, y),Φ2(x, y) ¦
µ1(x, y)P1(x, y)dx+ µ1(x, y)Q1(x, y)dy = dΦ1(x, y),
µ2(x, y)P2(x, y)dx+ µ2(x, y)Q2(x, y)dy = dΦ2(x, y),
· g1(·), g2(·)¦ µ1g1(Φ1) = µ2g2(Φ2). K µ(x, y) := µ1g1(Φ1)´ (1.2.1)
È©Ïf.
y: N´y, lÑ.
~K: ¦)§ (1.2.8).
): Äk x = 0 Ú y = 0 Ñ´§ (1.2.8) ).
x 6= 0, y 6= 0 , § y3dx− 2xy2dy = 0 kÈ©ÏfÚÏÈ©©O
µ1(x, y) = y−5, Φ1(x, y) = xy−2.
§ 2x2dy = 0 kÈ©ÏfÚÏÈ©©O
µ2(x, y) = x−2, Φ2(x, y) = y.
g1(z) = z−2, g2(z) = z−1, =§ (1.2.8) È©Ïf
µ(x, y) = µ1g1(Φ1) = µ2g2(Φ2) = x−2y−1,
ÚÏÈ©
Φ(x, y) = 2 ln |y| − x−1y2.
14
1Ù ~©§Ä:£
§1.2.3 Aa=zT§§
1. Cþ©l§
/X
P1(x)P2(y)dx+Q1(x)Q2(y)dy = 0, (1.2.9)
§¡Cþ©l§. ´ Q1(x) ": x = x0 Ú P2(y) ": y = y0 Ñ´§
(1.2.9) ). éu Q1(x)P2(y) 6= 0, Cþ©l§ (1.2.9) duT§
P1(x)
Q1(x)dx+
Q2(y)
P2(y)dy = 0.
Ï Q1(x)P2(y) 6= 0, § (1.2.9) kÏÈ©∫P1(x)
Q1(x)dx+
∫Q2(y)
P2(y)dy = c,
Ù¥ c ´?¿~ê.
3 §1.1 ~ 4 ¥, ЯK y′(x) = y13 , y(1) = 0 )Ø. e¡|^©lCþ
§¦)Ñùa§Ð¯K)3¿^.
~K: f(y) 3 |y − a| ≤ σ þëY, y = a ´ f(y) ":. K§
y′(x) = f(y), (1.2.10)
l y = a þz:Ñu)Ñ¿^´∣∣∣∣∫ a±σ
a
dy
f(y)
∣∣∣∣ =∞.
y: 75. y. e ∣∣∣∣∫ a±σ
a
dy
f(y)
∣∣∣∣ <∞. (x0, y0) ´ 0 < |y − a| < σ ¥?:, Ø y0 > 0. d Peano ½n, § (1.2.10)
L (x0, y0) k), φ(x) ´LT:). Ï y = a ´ f(y) ":, Ø
y ∈ (a, a+σ]k f(y) > 0. ¤±X xl x0~, φ(x)~ (Ï φ′(x) = f(φ(x)) > 0).
duL y = a þ?:), ¤±3 x l x0 ~L§¥©ªk φ(x) > a. )
φ(x) 1)3«m (−∞, x0]. - b = limx→−∞
φ(x), Kk b ≥ a. l
∞ >
∫ y0
b
dy
f(y)=
∫ x0
−∞dx =∞.
15
§1.2 ÐÈ©
ùgñ`²∣∣∣∫ a±σa
dyf(y)
∣∣∣ =∞.
¿©5. y. b§ (1.2.10) L,: (x0, a) k,), P y = ψ(x), x ∈ J , Ù¥
J ´¹ x0 m«m. Ï y = ψ(x) Ú y = a ´§ (1.2.10) üØÓ), ¤±3
x1 ∈ J ¦ y1 := ψ(x1) ∈ (a− σ, a+ σ) \ a. ùÒk
∞ =
∣∣∣∣∫ y1
a
dy
f(y)
∣∣∣∣ =
∣∣∣∣∫ x1
x0
dx
∣∣∣∣ <∞.ùgñ`²§ (1.2.10) L y = a þ?:)Ñ´. y..
2. àg§
§
dy
dx= R
(yx
), (1.2.11)
Ú§
P (x, y)dx+Q(x, y)dy = 0, P,Q ´ m gàg¼ê, (1.2.12)
Ñ¡àg§. ¼ê P (x, y)¡ m gàg¼ê,XJéu?¿ s > 0Ñk P (sx, sy) =
smP (x, y).
- y = ux, àg§ (1.2.11) Ú (1.2.12) ©OzCþ©l§
xdu
dx= R(u)− u.
Ú
xm(P (1, u) + uQ(1, u))dx+ xm+1Q(1, u)du = 0.
ÏLþãCþ©l§¦)§ (1.2.11) Ú (1.2.12)¦).
~Kµ¦)§
xdy
dx− y = x tan
y
x. (1.2.13)
)µ- y = xu. K§ (1.2.13) =z
xdu
dx= tanu,
ÙÏ)
sinu = cx, Ù¥ c ´?¿~ê
16
1Ù ~©§Ä:£
¤±, § (1.2.13) Ï)
y = xu = x arcsin(cx),
Ù¥ c ´?¿~ê"
3. ©ª5§ª
/X
dy
dx= f
(ax+ by + c
px+ qy + r
), (1.2.14)
§¡©ª5§ª. §±ÏL·C=zCþ©l§. ¯¢þ
• c = r = 0 , (1.2.14) w,´àg§.
• ∆ := aq − bp = 0 , Ø a = λp, b = λq. - u = px+ qy, K§ (1.2.14) z
Cþ©l§
du
dx= p+ qf
(λu+ c
u+ r
).
• ∆ 6= 0 , (ξ, η) ´5§| ax + by + c = 0, px + qy + r = 0 ). -
x = u+ ξ, y = w + η, K§ (1.2.14) zàg§
dw
du= f
(au+ bw
pu+ qw
).
~Kµ¦)§
dy
dx=−x3 + xy2 + x
x2y + y3 + 3y. (1.2.15)
)µ- u = x2, v = y2, § (1.2.15) =z
dv
du=−u+ v + 1
u+ v + 3. (1.2.16)
ê§
−u+ v + 1 = 0, u+ v + 3 = 0
k) u0 = −1, v0 = −2. -
u = ξ − 1, v = η − 2
17
§1.2 ÐÈ©
§ (1.2.16) =z
dη
dξ=−ξ + η
ξ + η.
- η = wξ. þã§z
ξdw
dξ= −w
2 + 1
w + 1
ù´Cþ©l§, ¦)
ln[(w2 + 1)ξ2] + 2 arctanw = c, Ù¥ c ´?¿~ê.
£Cþ, § (1.2.15) ÏÈ©
ln[(x2 + 1)2 + (y2 + 2)2
]+ 2 arctan
y2 + 2
x2 + 1= c,
Ù¥ c ´?¿~ê.
4. Riccati §
/X
dy
dx= p(x)y2 + q(x)y + r(x), p(x) 6≡ 0, r(x) 6≡ 0, (1.2.17)
§¡ Riccati §. 5`, Riccati §ØU^ÐÈ©¦).
·K7. e® Riccati § (1.2.17) ), K§±^ÐÈ©¦).
y: y = φ(x) ´§ (1.2.17) ). - y = φ(x) + u, K§ (1.2.17) =z
du
dx= (2p(x)φ(x) + q(x))u+ p(x)u2,
ù´ Bernoulli §, Ï ±^ÐÈ©¦). y..
e¡(ØÑa Riccati §^ÐÈ©¦)¿^.
·K8. Riccati §
dy
dx= ay2 + bxm, a 6= 0, b,m ∈ R,
^ÐÈ©¦)=
m = 0, −2, − 4k
2k + 1, − 4k
2k − 1, k ∈ N.
18
1Ù ~©§Ä:£
y: ¿©5´ Daniel Bernoulli u 1725 cy², ¯¢þ§±^ÐÈ©ØJy². 7
5´ Joseph Liouville u 1841 cy². [y²lÑ.
5: Louville (Ø¿Â3u«µU^ÐÈ©¦)©§´~.
F"ÏL¦)ݺ¤k©§£ã¢S¯KÄåÆ ´Ø¢S, ÏdI&
¦#ÚnØ.ùÒíÄ©§ynØ),X Henry PoincareMá©
§½5nØ"
5. AÏC
k§vkÚ?n, IäN¯KäNé. e¡ÞAü~f.
1. §
dy
dx=f(x+ y2)
y,
ÏLC u = x+ y2 ±z¤Cþ©l§
du
dx= 1 + 2f(u).
2. §
dy
dx= sin(x+ y + 1),
ÏLC u = x+ y + 1 ±z¤Cþ©l§
du
dx= 1 + sinu.
3. §
(x− 2 sin y + 5)dx+ (2x− 3 sin y + 1) cos y dy = 0,
ÏLC u = sin y z¤
(x− 2u+ 5)dx+ (2x− 3u+ 1)du = 0.
§äk (1.2.14) /ª, Ï ¦).
19
§1.2 ÐÈ©
§1.2.4 5©§~êC´
!2g£5©§,8´ùã¦)5©§~k^µ~
êC´. T3±¦)§|ék^. Ä5©§
dy
dx+ p(x)y = q(x), (1.2.18)
Ù¥ p(x), q(x) 3m«m (α, β) þëY. q(x) ≡ 0 , ¡ (1.2.18) àg5§.
q(x) 6≡ 0 , ¡ (1.2.18) àg5§.
5©§3¢S)¹¥þ/^, 'X RL £´>6§
LdI
dt(t) +RI(t) = E(t),
Ò´5©§, Ù¥ L ´>a, R ´>, I(t) ´>´>6, E(t) ´> >Ø.
e¡0~êC´. Äk|^Cþ©l¦àg5§
dy
dx+ p(x)y = 0, (1.2.19)
Ï)
y = ce−∫p(x)dx,Ù¥ c ´?¿~ê.
Ùgòàg5§Ï)¥?¿~ê c ¤'u x ¼ê c(x). ò¼ê
y = c(x)e−∫p(x)dx,
\§ (1.2.18), ¿z
c′(x)e−∫p(x)dx = q(x).
¤±
c(x) =
∫q(x)e
∫p(x)dxdx+ c,
Ù¥ c ´?¿~ê. àg5©§ (1.2.18) Ï)
y = e−∫p(x)dx
(c+
∫q(x)e
∫p(x)dxdx
),
Ù¥ c ´?¿~ê.
N5:
20
1Ù ~©§Ä:£
• éu x0 ∈ (α, β), 5©§ (1.2.18) ÷vЩ^ y(x0) = y0 )
y = e−
∫ xx0p(s)ds
(y0 +
∫ x
x0
q(t)e∫ tx0p(s)ds
dt
), x ∈ (α, β).
• 5§ (1.2.18) Ï)^½È©5L«
y = e−
∫ xx0p(s)ds
(c+
∫ x
x0
q(t)e∫ tx0p(s)ds
dt
)= c e
−∫ xx0p(s)ds
+
∫ x
x0
q(t)e∫ txp(s)dsdt,
Ù¥ x0 ∈ (α, β) ´?¿½:, c ´?¿~ê.
dþãÏ)ÚЯK)Lª, N´
·K9. 5©§)äke5µ
• àg5§ (1.2.19) )½öðu"½öðØu";
• 5§ (1.2.18) )3 p(x), q(x) ëY«m (α, β) þ3ëY;
• àg5§ (1.2.19) )?¿5|ÜE´ (1.2.19) );
• àg5§ (1.2.19) )àg§ (1.2.18) )ÚE´ (1.2.18) );
• 5§ (1.2.18) ü)´ (1.2.19) );
• 5©§ (1.2.18) ЯK)3.
~K:
1. ¦5©§ xy′ + (1− x)y = e2x (0 < x <∞) ÷v limx→0+
y(x) = 1 ).
) : §±¤
y′ =x− 1
xy +
1
xe2x.
§kÏ)
y = x−1ex(c+ ex),
Ù¥ c ´?¿~ê. Ï limx→0+
ex(c+ ex) = c+ 1, ¦ x→ 0+ , y(x) 4
3, 7Lk c = −1. q
limx→0+
ex(ex − 1)
x= 1.
21
§1.2 ÐÈ©
¤±§÷v½^)
y(x) = x−1ex(ex − 1).
2. a > 0, f(x) ´ëY 2π ±Ï¼ê, ¦©§
dy
dx+ ay = f(x), (1.2.20)
±Ï).
) : § (1.2.20) Ï)
y(x) = ce−ax +
∫ x
0
f(s)ea(s−x)ds. (1.2.21)
Äky² y(x) ´ 2π ±Ï) ⇐⇒ y(2π) = y(0). 75´w,. ey¿©5. -
z(x) = y(x+ 2π). K
dz
dx(x) =
dy
dx(x+ 2π) = −ay(x+ 2π) + f(x+ 2π) = −az(x) + f(x).
ùÒy² z(x) ´§ (1.2.20) ). q z(0) = y(2π) = y(0), d5§Ð¯K
)5 y(x+ 2π) = z(x) = y(x).
dÏ)Lª, l y(2π) = y(0) )
c =1
e2aπ − 1
∫ 2π
0
f(s)easds.
§ (1.2.20) k±Ï)
y(x) =1
e2aπ − 1
(∫ 2π
0
f(s)ea(s−x)ds+
∫ x
0
f(s)ea(2π+s−x)ds−∫ x
0
f(s)ea(s−x)ds
)=
1
e2aπ − 1
∫ x+2π
x
f(s)ea(s−x)ds.
3. f(x) 3 [0,∞) þëY, limx→∞
f(x) = b ∈ R. ¦y
(a) a > 0 , § (1.2.20) ¤k) x→∞ 4Ñ´ ba .
(b) a < 0 , § (1.2.20) k) x→∞ 4´ ba .
y : (a) d§ (1.2.20) Ï)Lª (1.2.21) , § (1.2.20) ?) y(x) 3 [0,∞)
þëY. ¤±k
limx→∞
y(x) = limx→∞
eaxy(x)
eax= limx→∞
y′(x) + ay(x)
a= limx→∞
f(x)
a=b
a,
22
1Ù ~©§Ä:£
Ù¥1ª^ Hospital K, 1nª^ y(x) ´§ (1.2.20) ).
(b) a < 0 , dbÈ©∫∞
0f(s)easds Âñ. 3Ï) (1.2.21) ¥, -
c = c0 −∫ ∞
0
f(s)easds.
KÏ)±¤
y(x) = e−ax(c0 +
∫ x
∞f(s)easds
).
¤±d Hospital K
limx→∞
y(x) = limx→∞
c0 +∫ x∞ f(s)easds
eax=
b
a, c0 = 0,
∞ (−∞), c0 6= 0.
¤± a < 0§ (1.2.20)k) y(x) =∫ x∞ f(s)ea(s−x)ds x→∞4´ b
a.
§1.2.5 Ûª§
!ÄÛª§
F
(x, y,
dy
dx
)= 0. (1.2.22)
¦).
1. y )ѧ
§ (1.2.22) ¤
y = f(x, p), p =dy
dx, (1.2.23)
Ù¥ f(x, p)ëY.ò p#ÏCþ, é§ (1.2.23)ü>'u x¦, p'u
x êwª©§
(fx(x, p)− p)dx+ fp(x, p)dp = 0. (1.2.24)
• e§ (1.2.24) kÏ) p = u(x, c), K§ (1.2.23) kÏ) y = f(x, u(x, c)), Ù¥ c ´
?¿~ê. éA)kaq(Ø. 5¿: ØUé y′(x) = u(x, c) ¦È© y, Ïù
y ؽ´§). e¡ Clairaut §~fò`²ù:.
23
§1.2 ÐÈ©
• e§ (1.2.24) kÏ) x = v(p, c), K§ (1.2.23) k¹ëê p Ï)
x = v(p, c),
y = f(v(p, c), p),
Ù¥ c ´?¿~ê. éA)kaq(Ø.
~K:
1. ¦) Clairaut §
y = xp+ f(p), p =dy
dx, f ′′(p) 6= 0.
): é Clairaut §ü>'u x ¦
(x+ f ′(p))dp
dx= 0.
d dpdx = 0 Clairaut §Ï)
y = cx+ f(c),
Ù¥ c ´?¿~ê. 5¿, Clairaut §Ï)dx¤.
d x+ f ′(p) = 0 Clairaut §A)
x = −f ′(p), y = xp+ f(p),
Ù¥ p ´ëê.
?Ú/, Ï f ′′(p) 6= 0, $^Û¼ê3½nl x = −f ′(p) )Ñ p = ω(x). K Clairaut
§A)¤
y = xω(x) + f(ω(x)). (1.2.25)
N5:
• Clairaut §´IêÆ[ Alexis Clairaut (1713–1765) ß 1734 cÚ\.
• L Clairaut§A)þ?:´Ï)¥^.¯¢þ, (x0, y(x0))´A
) (1.2.25) þ?:, Kk y′(x0) = ω(x0). ¤±A)L (x0, y(x0)) §
y − y(x0) = ω(x0)(x− x0),
=
y = c0x+ f(c0), c0 = ω(x0).
24
1Ù ~©§Ä:£
• A) (1.2.25) ØU^Ï)L«, Ï ω(x) Ø´~ê. ¯¢þ, l x+ f ′(ω(x)) ≡ 0
ω′(x) = − 1
f ′′(ω(x))6= 0.
• þãü:`² Clairaut §A)éAÈ©þ?:ÑkÏ)¥^È©
ÏL, üö3T:.
2. ¦)§
y = p2 − xp+1
2x2, p = y′(x).
): éþã§ü>'u x ¦
(2p− x)
(1− dp
dx
)= 0.
|^T§A)ÚÏ)§A)ÚÏ)
y =1
4x2, y =
1
2x2 + cx+ c2,
Ù¥ c ´?¿~ê.
w,, A)ÚÏ)¥z^È©Ñ´Ô. N´y, Ï)¥z^ÔÑA
)k:, Ø:Ï)È© uA)È©þ.
dþãü~f¥A)¤äkÓ5Ú\Vg.
y = φ(x), x ∈ J ´Ûª§ (1.2.22) ), Γ ´T)éAÈ©. e
∀ q ∈ Γ, § (1.2.22) Ñk,^È©3 q : Γ , ¡ y = φ(x), x ∈ J ´§
(1.2.22) Û).
2. Øw¹gCþ x ½ÏCþ y §
éuØw¹gCþ x §
F (y, p) = 0, p =dy
dx. (1.2.26)
Ï F (y, p) = 0 L« (y, p) ²¡eZ^,
y = g(t), p = h(t), t ∈ J,
´ (1.2.26) ëêzL«, g(t), h(t) ëY, h(t) 6= 0. Kk
dx =1
pdy =
g′(t)
h(t)dt.
25
§1.2 ÐÈ©
éþªÈ©§ (1.2.26) ¹ëê t Ï)
x =
∫g′(t)
h(t)dt+ c, y = g(t),
Ù¥ c ´?¿~ê.
éuØw¹ÏCþ y §
F (x, p) = 0, p =dy
dx. (1.2.27)
x = g(t), p = h(t), t ∈ J,
´ F (x, p) = 0 3 (x, p) ²¡ëêzL«, g(t), h(t) ëY. Kk
dy = pdx = g′(t)h(t)dt.
éþªÈ©§ (1.2.27) ¹ëê t Ï)
x = g(t), y =
∫g′(t)h(t)dt+ c,
Ù¥ c ´?¿~ê.
5: )/X (1.2.26) Ú (1.2.27) §'´XÛëêzùa§.
~K: ¦)e§
1) (y′)3 − y2(a− y′) = 0, a 6= 0;
2) ey′ − y′ = x.
): 1) - y′ = p, y = pt. §z
p2(p− t2(a− p)) = 0.
= p = 0 ½ p = at2
1+t2 . dcö§A) y = 0. dö
y =at3
1 + t2.
¤±
dx =1
pdy =
3 + t2
1 + t2dt.
26
1Ù ~©§Ä:£
È©§¹ëê t Ï)
x = t+ 2 arctan t+ c, y =at3
1 + t2,
Ù¥ c ´?¿~ê.
2) - y′ = p, ¿± p ëê x = ep − p. k
dy = pdx = p(ep − 1)dp.
È©§¹ëê p Ï)
x = ep − p, y = pep − ep − 1
2p2 + c,
Ù¥ c ´?¿~ê.
§1.2.6 p©§
1. Øw¹gCþ n ©§
F
(y,dy
dx, . . . ,
dny
dxn
)= 0, (1.2.28)
¡g£©§. n = 1 , § (1.2.28) ¦)®²ùL. n > 1 , ±ÏL
é§ (1.2.28) ü5¦).
- z = y′(x), ¿ò y À#gCþ. Kk
d2y
dx2= z
dz
dy,
d3y
dx3= z
(dz
dy
)2
+ z2 d2z
dy2, . . .
n § (1.2.28) z± z ÏCþ, y gCþ n− 1 §
G(y, z, z′(y), . . . , z(n−1)(y)
)= 0. (1.2.29)
XJ§ (1.2.29) k) z = ψ(y), KÏL¦) y′(x) = z = ψ(y) § (1.2.28) ).
~K:
1. ¦)§ 2yy′′ = (y′)2 + y2.
): - z = y′(x), §z
2yzdy
dy= z2 + y2, = y
dw
dy= w + y2,
27
§1.2 ÐÈ©
Ù¥ w = z2. w, y = 0 ´§). y 6= 0 , þ㧴5§, N´§
Ï)
w = cy + y2,
Ù¥ c ´?¿~ê.
du y = −c 6= 0 Ø´§), )§
y′(x) = z = ±√w = ±
√cy + y2,
§Ï)
ln |y +c
2+√cy + y2| = ±x+ c1,
Ù¥ c, c1 ´?¿~ê.
2. ¦)êƧ
x′′(t) = −a2 sinx, Ù¥ a2 = g/l.
): - v = x′(t), KlêƧ
v2 = 2a2 cosx− c,
Ù¥ c ´·~ê. k
x′(t) = ±√
2a2 cosx− c, (1.2.30)
= ∫dx√
2a2 cosx− c= ±
∫dt.
>´ýÈ©, Ã^мêL«Ñ5.
|x| 1 , sinx ≈ x. 3êƧ¥^ x O sinx
v2 = −a2x2 + c2,
=
x′(t) = ±√c2 − a2x2.
È©
x = A sin(at+D),
Ù¥ A ´Ì, D ´Ð© . ù´±Ï 2π/a Ä.
28
1Ù ~©§Ä:£
²L², Ìé (0 < A < π/6) , x = A sin(at + D) O(/£ã¨$
Ä.
Ì, x = A sin(at + D) ãã¨$Ä. I?Úïħ (1.2.30).
du x = A x′ = 0, ¤± c = 2a2 cosA. ü$ıÏ
T (A) = 4
∫ A
0
dx√2a√
cosx− cosA=
2√
2
a
∫ 1
0
Adu√cosAu− cosA
.
?Ú±y²
limA→0
T (A) =2π
a, lim
A→πT (A) =∞.
cö´$ıÏ.
2. "$ê§
F (x, y(k)(x), y(k+1)(x), . . . , y(n)(x)) = 0,
3C z = yk(x) e=z n− k §
F (x, z, z′, . . . , z(n−k)) = 0.
XJ¦Ñù n− k §)
z = φ(x, c1, . . . , cn−k),
ÏL k gÈ©Ò±§).
~K: ¦e§)
1. (y′′′)2 + x2 = 1, 2. x2y′′ = (y′)2.
): 1. - z = y′′′, Kk
z2 + x2 = 1.
Tê§k¹ëê)
z = cos t, x = sin t.
é
y′′′ = z = cos t,
29
§1.2 ÐÈ©
È©
y = − sin t+ c1t2 + c2t+ c3.
¤±§¹ëê t Ï)
x = sin t, y = − sin t+ c1t2 + c2t+ c3,
Ù¥ c1, c2, c3 ´?¿~ê.
2. - z = y′. §=z
x2z′ = z2.
È©
z =x
1 + cx, Ù¥ c ´?¿~ê.
é§
y′ =x
1 + cx,
驤)
y(x) =
1
2x2 + c1, c = 0,
x
c− 1
c2ln |1 + cx|+ c2, c 6= 0,
Ù¥ c1, c ´?¿~ê.
§1.2.7 Mathematica ¦)~©§
!0XÛ^Mathematica¦)~©§. Mathematica©a.´Math-
ematica Notebook, ©¶´ *.nb. |^ Mathematica '´Æ¬^èü¥ Help e
Find Selected Functions,l¥é¤I$.UìA$KÑ\,2ÏL Shift+Enter
$1O. e¡0Aü~f5`².
~K:
1. ^ Mathematica ¦§
y′ + 5xy = x3,
Ï).
): 1ÚÑ\§
DSolve[y′[x] + 5x y[x] == x∧3, y[x], x]
30
1Ù ~©§Ä:£
5¿¦Èi1mk.
1ÚU Shift +Enter =ÑÑ(J
y[x]− > 1
25(−2 + 5x2) + e−
5x2
2 C[1]
5¿, Ù¥ C[1] ´?¿~ê. ùÒ§Ï).
2. ¦§
y′′(x) + ay′(x) + by(x) = 3,
Ï).
): 1ÚÑ\§
DSolve[y′′[x] + a y′[x] + b y[x] == 3, y[x], x]
1ÚU Shift +Enter =ÑÑ(J
y[x]− > 3b + e
12 (−a−
√a2−4b)xC[1] + e
12 (−a+
√a2−4b)xC[2]
5¿, Ù¥ C[1], C[2] ´?¿~ê. ùÒ§Ï).
3. ^ Mathematica ¦)ЯK
y′(x) = ay(x), y(0) = 1,
): Ñ\§
DSolve[y′[x] == a y[x], y[0] == 1, y[x], x]
$1ÑÑ(J
y[x]− > eax
4. ¦)ЯK
y′(x) = cosx sec(2y), y(π/4) = 0.
): Ñ\§
DSolve[y′[x] == Cos[x]Sec[2 y[x]], y[Pi/4] == 0, y[x], x]
$1ÑÑ(J
y[x]− > 12ArcSin[2(− 1√
2+ Sin[x])]
5. ¦)©§
(y′(x))2 + y2 = 1.
): Ñ\§
31
§1.3 SK
DSolve[(y′[x])∧2 + (y[x])∧2 == 1, y[x], x]
$1ÑÑ(J
y[x]− > 1− 2Sin[ 12 (−x− iC[1])]2, y[x]− > 1− 2Sin[ 1
2 (x− iC[1])]2
Ù¥ i ´Jêü . dÑüÏ).
§1.3 SK
1. Ñe~©§ê.
1.
(xyd2y
dx2
)3
+ cos y5 = sec(xy), 2. tan
(dy
dx
)+ x2y2 = 1.
2. y½¼ê´A~©§)½Ï), ½Ð¯K).
2.1. ¼ê y(x) = 12 (x√
1− x2 + arcsinx) ´§(dy
dx
)2
+ x2 = 1,
);
2.2. ¼ê x = φ(t, c1, c2, c3) = c1e−3t + c2e
t cos(2t) + c3et sin(2t), c1, c2, c3 ∈ R ~ê, ´
§
x′′′(t) + x′′(t)− x′(t) + 15x(t) = 0,
3 t ∈ R þÏ);
2.3. ¼ê x = φ(t, c1, c2) = c1e−t + c2te
−t + et
4 + 12 sinx, c1, c2 ∈ R ~ê, ´§
x′′(t) + 2x′(t) + x(t) = et + cos t,
3 t ∈ R þÏ);
2.4. ¼ê
y(x) = e∫ xx0a(s)ds
(y0 +
∫ x
x0
b(t)e−
∫ tx0a(s)ds
dt
).
´Ð¯K
dy
dx= a(x)y + b(x), y(x0) = y0,
3 x ∈ (α, β) þ), Ù¥ a(x), b(x) 3 (α, β) ⊂ R þëY, x0 ∈ (α, β), y0 ∈ R.
32
1Ù ~©§Ä:£
3. ¼ê f(x, y) 3m« Ω ⊂ R2 þëY, 'u y ÷vÛÜ Lipschitz ^. Áyé?
¿k.48 D ⊂ Ω, f(x, y) 3 D þ'u y ÷v Lipschitz ^, = ∃L > 0 ¦
|f(x, y1)− f(x, y2)| ≤ L|y1 − y2|, ∀ (x, y1), (x, y2) ∈ D.
4. ¼ê f(x, y) 3m« Ω ⊂ R2 þ'u y êëY. Áy f(x, y) 3 Ω þ'u y
÷vÛÜ Lipschitz ^.
5. y²Ð¯K
y′(x) = f(x, y), y(0) = y0, y0 ∈ R ´?¿½~ê,
)Ø3, Ù¥
f(x, y) =
−1, |x| ≥ 1,
(−1)n, 1n+1 ≤ |x| <
1n , n ∈ N,
0, x = 0.
6. ©§ï
6.1. þ m :ë3üཥ:. 5Xê k > 0.
ò:÷X.l¥%, tm¦:$Ä÷v©§.
6.2. &ì1º´^=¡, ¦l:1 u1å²&ì1º
¤²11åÑ, Á¦&ì1º÷v©§.
6.3. ÓElôHW A :Ñu ué¡W B :Ê1, 3Ê1L§¥
EÞ©ª B :. bà° L, Y6Ý v, E3·YÊ1Ý V
(V > v). Á¦E$1÷v©§.
7. ^©½È©Ïf¦)e§
7.1. (x3 + xy2)dx+ (x2y + y3)dy = 0;
7.2. (y cosx− x sinx)dx+ (y sinx+ x cosx)dy = 0;
7.3. (x2 + y2 − 1)dx− 2xydy = 0;
7.4. 2xy ln ydx+ x2dy = 0;
33
§1.3 SK
7.5. 2xy ln ydx+ (x2 + y2√
1 + y2)dy = 0;
7.6. 2xy3dx+ (x2y2 − 1)dy = 0;
7.7. y2dx+ (2x2 − xy)dy = 0.
8. y²·K 6.
9. b¼ê P (x, y), Q(x, y), µ1(x, y), µ2(x, y) 3,²¡« Ω þëY. e µ1, µ2 ´
§
P (x, y)dx+Q(x, y)dy = 0, (1.3.1)
3 Ω ¥üÈ©Ïf, µ1/µ2 6=~ê. Áy µ1/µ2 = c (c ~ê)´§ (1.3.1)
ÏÈ©.
10. ¦)e§
10.1. xy′ + 1 = ey;
10.2. y′ sinx sin y − 2 cosx cos3 y = 0;
10.3.∫ x
1
√1 + (y′(s))2ds = 3x
13 + y(x);
10.4. y′(x) = eyx + y
x ;
10.5. y′ + 3x2 =√x3 + y;
10.6. y3dx− 2(xy2 − x2)dy = 0;
10.7. (sinx− 2y + 3) cosxdx+ (2 sinx− 4y − 3)dy = 0;
10.8. y′ = y/(2y ln y + y − x);
10.9. y′ − y tanx = secx, y(0) = 0;
10.10. x2y′ + 2(xy − 3)2 = 0.
11. ѦЯK
y′(x) = yk, y(0) = 0, k ≥ 0
k)¤k k, ¿éù k ¦ÑA).
34
1Ù ~©§Ä:£
12. b¼ê f(x) 3 R ,mf8þëY, f ′(0) 3, ÷v¼ê§
f(x+ y) = 2f(x)− f(x)f(y).
Áy²¼ê f(x) 3 R þëY, ¿¦¼ê f(x).
13. b¼ê f(x) 3(−π2 ,
π2
)þëY, f ′(0) 3, ÷v¼ê§
f(x+ y) =f(x) + f(y)
1− f(x)f(y).
Á¦¼ê f(x).
14. Áy²·K 8 ¿©5.
15. b¼ê p(x), q(x) 3 R þëY, y1(x), y2(x) ´5©§
y′ + p(x)y = q(x),
3 R þüØÓ). ÁyéuT§3 R þ?) y(x) Ñk
y(x)− y1(x)
y2(x)− y1(x)= c, x ∈ R,
Ù¥ c ´~ê.
16. b¼ê q(x) 3 R þëY, limx→0
q(x) = b, a ∈ R ´~ê. Áy5©§
xy′ + ay = q(x),
(a) a > 0 , k) x→ 0 43,
(b) a < 0 , ¤k) x→ 0 4Ñ3, 4Ó.
17. b¼ê q(x) 3 R þëY, k.. Áy5©§
y′ + y = q(x),
(a) 3 R þkk.), ¿¦ÑT);
(b) XJ q(x) ± ω ±Ï, KTk.)± ω ±Ï.
18. ®¼ê f(x) ÷v ∫ x
0
f(t)
f2(t) + tdt = f(x)− 1.
Á¦¼ê f(x).
35
§1.3 SK
19. y²§
xy′ − (2x2 + 1)y = x2,
k) x→∞ kk4, ¿¦ù)Ú4.
20. ¦SK 6 ¥ïán§).
21. ¦)eÛª§
21.1. y = 2xy′ + (y′)2;
21.2. y = xy′ + y′ − (y′)2;
21.3. (x2 + 1)(y′)2 = 1;
21.4. (x2 − 4)(y′)2 + 2xyy′ + y2 = 0;
21.5. x√
1 + (y′)2 − y′ = 0;
21.6. (y′)3 − x3(1− y′) = 0;
21.7. y(1 + (y′)2) = 4;
21.8. y2(y′ − 1) = (2− y′)2.
22. ¦)ep©§
22.1. yy′′ + (y′)2 = 1;
22.2. (y′)2 = 1 + yy′′;
22.3. y′′ + (y′)2 = 2e−y;
22.4. y′ = xy′′ + (y′′)2;
22.5. y′′ = 2y′ + x;
22.6. y′′ − ex(y′)2 = 0.
36
1Ù ©§)35Ú5
~©§Ð¯K)35y²kõ«,õêÖæ^´ Picard
Åg%C. Öò$^Cy²,|^ålm¥Ø Nny²
©§)35(Ø, dk0ùý£.
§2.1 ý£µålmØ Nn
§2.1.1 ålm
3È©¥·ÆL4Ú¼êëY5. ÃØ´?Ø4´ëY5,ÑIkå
l. ¢ê8 R ¥?¿ x, y ålÏ~½Â ρ(x, y) = |x− y|, Ù¥| · |L«R¥:ýé.
é Rn¥?¿ü: x = (x1, . . . , xn), y = (y1, . . . , yn), ρ(x, y) =√
(x1 − y1)2 + . . .+ (xn − yn)2
½Â x y mål. N´y, ¼ê ρ(·, ·) ÷ve5µé ∀x, y, z ∈ Rn k
ρ(x, y) ≥ 0; ρ(x, y) = ρ(y, x); ρ(x, y) ≤ ρ(x, z) + ρ(z, y).
$^ù5±3Ä8Üþ½Âål.
X´?¿8Ü. é∀x, y ∈ X, k¢ê ρ(x, y) éA, ÷v:
1) K5: ρ(x, y) ≥ 0§ρ(x, y) = 0⇐⇒ x = y;
2) é¡5: ρ(x, y) = ρ(y, x);
3) nت: é∀x, y, z ∈ Xk ρ(x, y) ≤ ρ(x, z) + ρ(z, y),
¡ ρ(x, y)x ymål. ¡ (X, ρ)± ρålålm. ±Bå§3ål
®^e§¡X´ålm. XJY ⊂ X§K (Y, ρ)´ålm, ¡X
fm.
~K:- C[a, b]L«½Â3 [a, b]þëY¼êN¤8Ü.é?¿x(t), y(t) ∈ C[a, b],
½Â
ρ(x, y) = maxt∈[a,b]
|x(t)− y(t)|. (2.1.1)
K ρ(x, y)´C[a, b]þål. l C[a, b]´± ρ(x, y)ålålm.
y: N´y ρ÷vål½Â¥K5Úé¡5. e¡y² ρ÷vnت.
37
§2.1 ý£µålmØ Nn
é∀x(t), y(t), z(t) ∈ C[a, b]k
|x(t)− y(t)| ≤ |x(t)− z(t)|+ |z(t)− y(t)|
≤ maxt∈[a,b]
|x(t)− z(t)|+ maxt∈[a,b]
|z(t)− y(t)| = ρ(x, z) + ρ(z, y).
k ρ(x, y) ≤ ρ(x, z) + ρ(z, y). ùÒy² ρ´C[a, b]þål. l (C[a, b], ρ) ´å
lm.
aquîªmÈ©nØ¥4½Â, ±3ålm¥½Â4Vg.
(X, ρ)´?ålm.
• : xn ⊂ XÂñ, XJ3 z ∈ X§¦é ∀ ε > 0§3N ∈ N, n ≥ Nk
ρ(xn, z) < ε.
d¡: xnÂñ z, ½ z´ xn4. P
limn→∞
xn = z, ½ xn −→ z(n→∞).
• : xn ⊂ X´ Cauchy , XJé ∀ ε > 0, ∃N ∈ N ¦ n,m ≥ Nk
ρ(xn, xm) < ε.
e¡?Øålm¥45.
·K10. (X, ρ)´?ålm, xn ⊂ X´:"
(a) XJ xnÂñ, KÙ4;
(b) XJ xnÂñ, KÙ?¿fS7Âñ;
(c) XJ xnÂñ, KÙ7 Cauchy .
y: (a). z, y´ xn4. Ké∀ ε > 0, ∃N ∈ N, ¦n > Nk
ρ(xn, z) <ε
2, ρ(xn, y) <
ε
2.
é?¿n0 > Nk
ρ(z, y) ≤ ρ(z, xn0) + ρ(xn0
, y) < ε.
38
1Ù ©§)35Ú5
d ε > 0?¿5 ρ(z, y) = 0§= z = y.
(b)Ú (c). d4½ÂÚ (a)y²N´. SK, ÖögC¤. y..
5µ¯¤±, îªm¥Âñ:=´ Cauchy . 3ålm¥, k~f
L² Cauchy 7Âñ.
ålm¡´, XJ§?¿ Cauchy ÑÂñ.
·K11. ρ ´ (2.1.1) ¥½Â C[a, b] þål, Kålm (C[a, b], ρ) ´.
y: xn ⊂ C[a, b] ´ Cauchy . Ké ∀ ε > 0, ∃N > 0 ¦ m,n > N
ρ(xn, xm) = maxt∈[a,b]
|xn(t)− xm(t)| < ε
2.
AO/k
|xn(t)− xm(t)| < ε
2, t ∈ [a, b].
de¡·K 13, ∃x(t)¦ xn(t) ⇒ x(t). ùp⇒L«Âñ. Ï xn(t) ∈ C[a, b],
de¡·K 14 , x(t) ∈ C[a, b]. ùÒy² C[a, b] ¥?¿ Cauchy ÑÂñ C[a, b]
¥:. Ï C[a, b] . y..
íØ12. K > 0 ´?½~ê. - C = x(t) ∈ C[a, b]; |x(t)| ≤ K, t ∈ [a, b], ρ ´
(2.1.1) ¥½Â C[a, b] þål, K (C, ρ) ´ålm.
·K13. E ⊂ R§fn(t) ´½Â3 E þ¼ê. K3 E þ¼ê f(t) ¦ fn(t) ⇒
f(t) [E] ⇐⇒ é ∀ ε > 0, ∃N > 0 ¦ n,m > N k |fm(t)− fn(t)| < ε, t ∈ E.
y: 75. é ∀ ε > 0. Ï fn(t) ⇒ f(t) [E], ∃N > 0 ¦ n > N
|fn(t)− f(t)| < ε
2, t ∈ E.
n,m > N k
|fm(t)− fn(t)| ≤ |fm(t)− f(t)|+ |f(t)− fn(t)| < ε, t ∈ E.
¿©5. é?¿½ t ∈ E, db fn(t) ⊂ R ´ Cauchy S, l Âñ. PÙ
4 f(t). K f(t) ´ E þ¼ê. ey fn(t) ⇒ f(t) [E].
é ∀ ε > 0. db, ∃N > 0, ¦ n > N , é¤k k ∈ N k
|fn+k(t)− fn(t)| ≤ ε
2, t ∈ E.
39
§2.1 ý£µålmØ Nn
3þª¥- k →∞
|f(t)− fn(t)| ≤ ε
2< ε, t ∈ E.
= fn(t) ⇒ f(t) [E]. y..
·K14. fn(t) ⊂ C(E), fn(t) ⇒ f(t) [E]. K f(t) ∈ C(E).
y: é ∀ t0 ∈ E, ∀ ε > 0. du fn(t) ⇒ f(t) [E], ∃N > 0, ¦ n > N ,
|fn(t)− f(t)| < ε
3, t ∈ E.
Ï fN+1(t) 3 E þëY, ∃ δ > 0 ¦ t ∈ E |t− t0| < δ
|fN+1(t)− fN+1(t0)| < ε
3.
l |t− t0| < δ
|f(t)− f(t0)| ≤ |f(t)− fN+1(t)|+ |fN+1(t)− fN+1(t0)|+ |fN+1(t0)− f(t0)| < ε.
ùÒy² f(t) 3 t0 ëY. d t0 ∈ E ?¿5, f(t) ∈ C(E). y..
N5:
1) 3 C[a, b] þ½Â
ρ∗(x, y) =
∫ b
a
|x(t)− y(t)|dt, ∀x(t), y(t) ∈ C[a, b],
K ρ∗ ´ C[a, b] þål. C[a, b] 3ål ρ∗ eØ´.
2) Ω ⊂ Rn ´k.4«. 3 C(Ω) þ½Â
ρ(y1, y2) = maxx∈Ω|y1(x)− y2(x)|, ∀ y1, y2,∈ C(Ω).
K (C(Ω), ρ) ´ålm.
§2.1.2 Ø Nn
Ø Nn3yêƯõ©|¥åX^. e!ò^Ø Nny²
©§)35.
(X, ρ) ´ålm, T : X → X ´N.
40
1Ù ©§)35Ú5
• e ∃α ∈ [0, 1) ¦é ∀x, y ∈ X k
ρ(Tx, Ty) ≤ αρ(x, y),
¡ T ´ X þØ N, α Ø ~ê.
• e ∃x0 ∈ X ¦ Tx0 = x0, ¡ x0 ´ T ØÄ:.
·K15. ålmþØ N´ëY"
y: öSdÖögC¤.
e¡Ø Nn, q¡ Banach ØÄ:½n, ´ålmnØ¥óä.
§´d Stefan Banach (1892õ1945,Å=êÆ[,y¼©ÛCÄ<)u 1922cÑ.
½n16. (Ø Nn) X ´ålm, T : X → X ´Ø N. K3
z ∈ X ¦ Tz = z, = T 3 X ¥kØÄ:.
y: α ´Ø N T Ø ~ê. Äky²ØÄ:35.
é ∀x0 ∈ X, -
xn = Tnx0 = T (Tn−1x0).
K xn ⊂ X ´ Cauchy . ¯¢þ, é ∀ ε > 0, N > 0 ¦
αN
1− αρ(x1, x0) < ε.
é ∀m ∈ N k
ρ(xm+1, xm) = ρ(Txm, Txm−1) ≤ αρ(xm, xm−1) = αρ(Txm−1, Txm−2)
≤ α2ρ(xm−1, xm−2) ≤ . . . ≤ αmρ(x1, x0).
n,m > N£Ø n > m¤
ρ(xm, xn) ≤ ρ(xm, xm+1) + ρ(xm+1, xm+2) + . . .+ ρ(xn−1, xn)
≤(αm + αm+1 + . . .+ αn−1
)ρ(x1, x0)
= αm1− αn−m
1− αρ(x1, x0) ≤ αm
1− αρ(x1, x0) < ε.
ùÒy² xn ´ X ¥ Cauchy .
41
§2.2 )35: Picard ½n
Ï X , ∃ z ∈ X ¦ xn → z. d xn = Txn−1 Ú T ëY5 Tz = z, = z ´
T ØÄ:.
5. y ∈ X ´ T ØÄ:, = Ty = y. K
ρ(y, z) = ρ(Ty, Tz) ≤ αρ(y, z).
Ï α < 1, ¤± ρ(y, z) = 0, = y = z. ùÒy²ØÄ:5.
N5µ½ny²Jø¦ålm X ¥Ø N T ØÄ:CqOÚØ
O. z ´ T ØÄ:, Ké ∀x0 ∈ X k
ρ(z, Tmx0) ≤ αm
1− αρ(Tx0, x0).
íØ17. (X, ρ) ´ålm. éN T : X → X, e3 α ∈ [0, 1) 9 n ∈ N ¦é
∀x, y ∈ X k
ρ(Tnx, Tny) ≤ αρ(x, y).
K T 3 X þ3ØÄ:.
y: öSÖögC¤.
N5µ'uØÄ:½nkéõ´LSN,Öö±3٧ƥºYÆ,ë [14, 49,
59].
§2.2 )35: Picard ½n
!$^Ø Nny²©§Ð¯K)35.
½n18. (Picard½n) f(x, y) 3 R2 ,m« Ω þëY, 'u y ÷vÛÜ Lipschitz
^. Ké ∀ (x0, y0) ∈ Ω, ©§Ð¯K:
dy
dx= f(x, y), y(x0) = y0, (2.2.1)
kL (x0, y0) ØòÿëY).
y: ´Ð¯K (2.2.1) duÈ©§
y(x) = y0 +
∫ x
x0
f(t, y(t))dt. (2.2.2)
42
1Ù ©§)35Ú5
Äky²È©§ (2.2.2) k½Â3 x0 ,þ).
Ý/«
R := (x, y); |x− x0| ≤ a, |y − y0| ≤ b ⊂ Ω,
Ù¥ a, b > 0. -
M = max(x,y)∈R
|f(x, y)|,
L ´ f(x, y) 3 R þ Lipschitz ~ê, =
|f(x, y1)− f(x, y2)| ≤ L|y1 − y2|, ∀ (x, y1), (x, y2) ∈ R.
δ ∈(0, min 1
L , a,bM ). Äålm (C, ρ), Ù¥
C = y(x) ∈ C[x0 − δ, x0 + δ]; |y(x)− y0| ≤ b.
½ÂN
(Ty)(x) = y0 +
∫ x
x0
f(t, y(t))dt, y(x) ∈ C.
K T ´ C þØ N. ¯¢þ, é ∀ y(x) ∈ C k (x, y(x)) ∈ R, x ∈ [x0 − δ, x0 + δ], ¤±d
y(x) Ú f(x, y) ëY5 (Ty)(x) ∈ C. qé ∀ y1(x), y2(x) ∈ C k
ρ(Ty1, T y2) = maxx∈[x0−δ,x0+δ]
∣∣∣∣∫ x
x0
(f(t, y1(t))− f(t, y2(t))) dt
∣∣∣∣≤ max
x∈[x0−δ,x0+δ]
∣∣∣∣∫ x
x0
|f(t, y1(t))− f(t, y2(t))| dt∣∣∣∣
≤ maxx∈[x0−δ,x0+δ]
∣∣∣∣∫ x
x0
L |y1(t)− y2(t)| dt∣∣∣∣
≤ L maxt∈[x0−δ,x0+δ]
|y1(t)− y2(t)| |x− x0| ≤ Lδρ(y1, y2).
T ´ C þØ N.
díØ 12 , C ´. ¤±dØ Nn, T 3 C þ3ØÄ:, P
φ0(x). =
φ0(x) = Tφ0(x) = y0 +
∫ x
x0
f(t, φ0(t))dt.
dþª, φ0(x) 3 [x0 − δ, x0 + δ] þëY. l φ0(x) ´Ð¯K (2.2.1) 3 [x0 −
δ, x0 + δ] þ).
Ùgy²þ¡Ð¯K (2.2.1) ÛÜ) φ0(x) ±òÿ3«mþ.
43
§2.2 )35: Picard ½n
Ï Ω ´m«, ¤± (x1, φ0(x1)) ∈ Ω, Ù¥ x1 = x0 + δ ½ x1 = x0 − δ. A^þã
ÓyЯK
y′(x) = f(x, y(x)), y(x1) = φ0(x1),
3,«m [x1 − δ1, x1 + δ1] k), P φ1(x), Ù¥ δ1 > 0.
dЯKÛÜ)5 φ0(x) = φ1(x), x ∈ [x0 − δ, x0 + δ] ∩ [x1 − δ1, x1 + δ1]. -
φ1(x) =
φ0(x), x ∈ [x0 − δ, x0 + δ],
φ1(x), x ∈ [x1 − δ1, x1 + δ1].
K φ1(x) ´Ð¯K (2.2.1) ½Â3 [x0 − δ, x0 + δ] ∪ [x1 − δ1, x1 + δ1] þ).
EþãL§±Ð¯K (2.2.1) 3) y = φ(x), x ∈ J , Ù¥ J ´
3«m. y..
N5:
1) Picard ½n¥« Ω ´XÛ(½? 5`¼ê f(x, y) Ѵмê, Ï Ω
Ò´¼ê f k½Â«.
2) ½Â3m« Ω þ©§²LÙ¥?:)3«mÑ´m«m. vk
½ny3«m. ~X©§
y′ = sec2 x,
Ï)´ y = tanx+ c. §?)3«mÝÑ´ π. ©§
y′ =1
1 + x2,
Ï)´ y = arctanx+ c. §?)3«mÑ´ R.
3) y = φ(x) ´Ð¯K (2.2.1) ), J = (α, β) ´T)3«m, K x → β− Ú
x→ α+ , (x, φ(x)) Ѫu«>. ∂Ω.
5¿: x→ β− ½ x→ α+ , φ(x) 4UØ3. ~X©§
y′ = − 1
x2cos
1
x,
½Â Ω = (R \ x = 0)× R. §)
y = sin1
x,
44
1Ù ©§)35Ú5
½Â3 (−∞, 0) ½ (0,∞) þ. x→ 0 , T)4Ø3.
4) Picard ½n²;y²´ÏLE Picard Sµ
φ0(x) = y0, x ∈ [x0 − δ, x0 + δ]
φ1(x) = y0 +∫ xx0f(t, φ0(s))ds, x ∈ [x0 − δ, x0 + δ]
φn(x) = y0 +∫ xx0f(t, φn−1(s))ds, n = 2, 3, . . . , x ∈ [x0 − δ, x0 + δ],
¿y² PicardS3 [x0− δ, x0 + δ]þÂñ55y²Ð¯K (2.2.1))
35Ú5. ë [16, 53, 55, 56].
4) ½n 18ÑЯK)3¿©^.,kéõ'u Picard½ní2,
3Ù§f^ey²Ð¯K (2.2.1))35. ë [13, 16, 24, 26].
§2.3 )35µPeano ½n
þ!Ñ~©§Ð¯K)3Ú5^. !òy²ëY5=y
©§Ð¯K)35. dkO.
!Arzela–Ascoli Ún
¼ê fn(x), x ∈ I ⊂ R ¡3 I þ
• k., XJ ∃K > 0 ¦é¤k n ∈ N, x ∈ I Ñk |fn(x)| ≤ K;
• ÝëY,XJé ∀ ε > 0Ñ ∃ δ > 0¦é¤k n ∈ N, x1, x2 ∈ I, |x1−x2| < δ
Òk |fn(x1)− fn(x2)| ≤ ε.
e¡(JѼê3Âñf¿©^.
Arzela–AscoliÚn. XJ¼ê fn(x)3k.48 I þk.!ÝëY,K fn(x)
kf3 I þÂñ.
Arzela–Ascoli Úny²A3z¼©ÛÖÑk (ë [59]), y²Ñ
´3Äm¥Ñ. ÖöBÖN¹ §6.1 ÑÐy².
N5:
1) Arzela–Ascoli Ún¥k.48 I ¤km«m, (ØE,¤á.
45
§2.3 )35µPeano ½n
2) Arzela–Ascoli Ún¥k.48 I ¤Ã¡«m, (Ø7¤á.
!)35: Peano ½n
½n19. f(x, y) 3m« Ω ⊂ R2 þëY, (x0, y0) ∈ Ω. KЯK (2.2.1) k
½Â33«mþëY).
y: l½n 18 y², IyЯK (2.2.1) 3 x0 ,S3ëY)=.
R := (x, y); |x− x0| ≤ a, |y − y0| ≤ b ⊂ Ω,
Ù¥ a, b > 0. -
M = max(x,y)∈R
|f(x, y)|, δ = min
a,
b
M
, I = [x0, x0 + δ].
XJ M = 0, = f(x, y) ≡ 0, (x, y) ∈ R, K y(x) = y0, x ∈ (x0− a, x0 + a), ´Ð¯K (2.2.1)
).
±eb M > 0. yЯK (2.2.1)3 I þ)35 (3 [x0− δ, x0] þ)
35aqy, lÑ).
1. 3 I þE Euler òS.
éz n, ò I ©¤ n °, P©: x0, x1, . . . , xn, Ù¥ xi < xj , 0 ≤ i < j ≤ n. 3
I þE Euler òXe: L (x0, y0) ã
ψ1(x) = y0 + f(x0, y0)(x− x0), x ∈ [x0, x1].
- y1 = ψ1(x1). L (x1, y1) ã
ψ2(x) = y1 + f(x1, y1)(x− x1), x ∈ [x1, x2].
d, 3z«m [xi, xi+1] þEã
ψi+1(x) = yi + f(xi, yi)(x− xi), x ∈ [xi, xi+1], yi = ψi(xi).
P γn ùã¿, ¡ I þ Euler ò.
2. Euler òLª.
46
1Ù ©§)35Ú5
P φn(x), x ∈ I Euler òLª, K
φn(x) = y0 +
∫ x
x0
f(t, φn(t))dt+ σn(x), x ∈ I, (2.3.1)
Ù¥3 Iþ σn(x) ⇒ 0 (n→∞). ¯¢þ,é ∀x ∈ I, ∃ s ∈ 0, 1, . . . , n−1¦ x ∈ (xs, xs+1],
l
φn(x) = y0 +
s−1∑k=0
f(xk, yk)(xk+1 − xk) + f(xs, ys)(x− xs).
Ï
f(xk, yk)(xk+1 − xk) =
∫ xk+1
xk
f(t, φn(t))dt+
∫ xk+1
xk
(f(xk, yk)− f(t, φn(t)))dt,
f(xs, ys)(x− xs) =
∫ x
xs
f(t, φn(t))dt+
∫ x
xs
(f(xs, ys)− f(t, φn(t)))dt,
¤±
σn(x) =s−1∑k=0
rn(k) + r∗n(k),
Ù¥
rn(k) =
∫ xk+1
xk
(f(xk, yk)− f(t, φn(t)))dt,
r∗n(s) =
∫ x
xs
(f(xs, ys)− f(t, φn(t)))dt, x ∈ (xs, xs+1].
ey σn(x) ⇒ 0, x ∈ I. Ï f 3 R þëY (f ´k.48 R þëY¼ê),
x ∈ [xk, xk+1] k
|x− xk| ≤δ
n, |φn(x)− yk| = |f(xk, yk)(x− xk)| ≤M(x− xk) ≤ Mδ
n,
¤±é ∀ k ∈ 0, 1, . . . , n− 1, ∀ ε > 0 Ñ ∃N > 0 ¦ n > N k
|f(xk, yk)− f(x, φn(x))| < ε
δ, x ∈ [xk, xk+1].
|rn(k)| <∫ xk+1
xk
ε
δdx =
ε
n, |r∗n(k)| <
∫ x
xs
ε
δdx ≤ ε
n, x ∈ (xs, xs+1].
l k |σn(x)| < ε, x ∈ I, = σn(x) ⇒ 0.
3. ¼êS φn(x) 3 I þk.ÝëY.
â Euler òEk γn ⊂ R, ¤± |φn(x) − y0| ≤ b, = |φn(x)| ≤ |y0| + b. l
φn(x) 3 I þk..
47
§2.3 )35µPeano ½n
eyÝëY. é ∀ ε > 0, δ = εM . é ∀ s, t ∈ I, ∀n ∈ N, Ïë: (s, φn(s)) Ú
(t, φn(t)) Ç0u −M Ú M m, ¤± |s− t| < δ Òk
|φn(s)− φn(t)| ≤M |s− t| < ε,
= φn(x) 3 I þÝëY.
4. 35y².
d Arzela–AscoliÚn,¼ê φn(x)3 I þ3Âñf,P φnk(x). -
φ(x) ´f4.
d (2.3.1) , φnk(x) ÷v
φnk(x) = y0 +
∫ x
x0
f(t, φnk(t))dt+ σnk(x), x ∈ I.
éþª- k →∞
φ(x) = y0 +
∫ x
x0
f(t, φ(t))dt, x ∈ I.
ùÒy² y = φ(x) ´Ð¯K (2.2.1) 3 I þ). y..
N5:
• éu½n 19 y²¥½ δ Ú I, ±9 I n + 1 ©: x0, x1, . . . , xn. E
¼êS φn(x) Xeµ
φn(x) =
yn0(x) := y0, x ∈ [x0, x1],
yn1(x) := y0 +∫ x− δ
n
x0f(s, yn0(s))ds, x ∈ (x1, x2],
yn2(x) := y0 +∫ x1
x0f(s, yn0(s))ds+
∫ x− δn
x1f(s, yn1(s))ds, x ∈ (x2, x3],
......
yn,n−1(x) := y0 +n−3∑i=0
∫ xi+1
xif(s, yni(s))ds
+∫ x− δ
n
xn−2f(s, yn,n−2(s))ds, x ∈ (xn−1, xn].
TS¡ Tonelli S. |^TSÚ Arzela–Ascoli Ún±y²½n 19.
• Peano ½nkéõ?Úí2, ë [20, 23] 9Ù¥ë©z.
3 Picard½nÚ Peano½n¥Ñ`²)½Â33«mþ. éuäN
§(½§)3«m¿ØN´.
48
1Ù ©§)35Ú5
~K:
1. y²§
y′ = (1− y2)ex2+y2 ,
z)3«mѴá«m.
): ϧmà¼ê f(x, y) = (1 − y2)ex2+y2 3 R2 þëY, Ïd§l R2 ¥?
:Ñu)Ñ3.
Äk´ y = φ0(x) := ±1 ´§). w,ùü)3«mÑ´ R. ?Ú/k
limx→±∞
φ0(x) = ±1.
é ∀ (x0, y0) ∈ R2 \ y = ±1, P y = φ(x) ´§L (x0, y0) ), )3«m´
J = (a, b). d)5, φ(x) φ0(x) 3 J þØ.
e¡Ò (x0, y0) ¤3ØÓ«©O?Ø.
(a) −1 < y0 < 1. Ï f(x, y) > 0, ¤±ok φ′(x) > 0, x ∈ J . Ï φ(x) 3 J þüNOk
.. -
limx→b−
φ(x) = β ≤ 1, limx→a+
φ(x) = α ≥ −1.
ey a = −∞, b =∞. é b =∞ ¹±y². a = −∞ ¹±aq/y²,
lÑ. y, b b < ∞. d)5, (b, β) 6∈ y = 1. Ï L (b, β) §k½Â3
,«m [b, b + σ) (σ > 0) þ), = φ(x) ½Â3 (a, b + σ) þ, J ´3«mgñ.
¤± φ(x) 3«m´ (−∞,∞).
?Ú/y β = 1, α = −1. Ï limx→∞
φ(x) = β, ¤± limx→∞
φ′(x) = 0 (ùd
L’Hospital KN´y). ò φ(x) \§¿- x → ∞ β = 1. Óny
α = −1.
(b) y0 > 1. Ï f(x, y) < 0,¤±ok φ′(x) < 0, x ∈ J . Ï φ(x)3 J þüN~ek.. Ó
þy² φ(x) m3«m [x0,∞), limx→∞
φ(x) = 1.
(c) y0 < −1. Ï f(x, y) < 0, ¤±ok φ′(x) < 0, x ∈ J . Ï φ(x) 3 J þüN~þk..
Óþy² φ(x) 3«m (−∞, x0], limx→−∞
φ(x) = −1.
y..
49
§2.4 )éÐÚëêëY65
§2.4 )éÐÚëêëY65
½©§ ¹këê,@o©§)Ùëêm'XXÛ?)qXÛ
6uЩ^? kwü~f. ЯK:
dx
dt= ax, x(t0) = x0,
) x(t) = x0ea(t−t0). §Ø=´gCþ t ¼ê, ´§ëê a ÚЩ^
(t0, x0) ¼ê. Ï)´§ëêÚЩ^¼ê, ±r) x(t) ¤ x(t, t0, x0, a).
ĹëꩧЯK:
dy
dx= f(x, y, λ), y(x0) = y0, (2.4.1)
Ù¥ λ ∈ K ⊂ Rm ´ m ëê. - t = x− x0, u = y − y0, KþãЯK=z
du
dt= f(t+ x0, u+ y0, λ), u(0) = 0.
=ЯK (2.4.1) Щ^=z¤#XÚëê. ÏdØ5, e¡Ä¹ë
êЯK:
dy
dx= f(x, y, λ), y(0) = 0. (2.4.2)
½n20. f(x, y, λ) 3« Ω×Λ ⊂ R2 ×Rm þëY, 'u y ÷vÛÜ Lipschitz ^,
=é ∀Z0 = (x0, y0, λ0) ∈ Ω× Λ, 3 Z0 3 Ω× Λ ¥ UZ0, 9~ê LZ0
¦
|f(x, y1, λ)− f(x, y2, λ)| ≤ LZ0 |y1 − y2|, ∀ (x, y1, λ), (x, y2, λ) ∈ UZ0 .
XJ (0, 0) ∈ Ω, KЯK (2.4.2) k), P y = φ(x, λ), φ(x, λ) 'uÙCþë
Y.
y: l½n 18 y², IyЯK (2.4.2) 3 x = 0 3). duëY´Û
Ü5, IyЯK)3 Λ ?:ëY=.
? λ0 ∈ Λ, 9± λ0 ¥%4¥ C0 ⊂ Λ. -
R = (x, y); |x| ≤ a, |y| ≤ b ⊂ Ω, M = maxR×C0
|f(x, y, λ)|.
db, f(x, y, λ) 3 R× C0 þk Lipschitz ~ê, P L.
50
1Ù ©§)35Ú5
δ ∈(0, min 1
L , a,bM ), I = [−δ, δ]. -
C = y(x, λ) ∈ C(I × C0); |y(x, λ)| ≤ b.
3 C þ½Âål
ρ(y1, y2) = max(x,λ)∈I×C0
|y1(x, λ)− y2(x, λ)|, ∀ y1, y2 ∈ C.
Kaqu·K 11 y² (C, ρ) ´ålm.
duЯK (2.4.2) duÈ©§
y(x, λ) =
∫ x
0
f(t, y(t, λ), λ)dt. (2.4.3)
½ÂN
(Ty)(x, λ) =
∫ x
0
f(t, y(t, λ), λ)dt, y(x, λ) ∈ C.
Kaqu½n 18 y²y, T ´ C þØ N. l dØ Nn, T 3 C þ
3ØÄ:, =ЯK (2.4.2) k) y = φ(x, λ), 3 I × C0 þëY.
aqu½n 18 y²Ð¯K (2.4.2) )½Â33«mþ, P J , T
)3 J × C0 þëY. y..
§2.5 SK
1. y²·K 10 5 (b) Ú (c).
2. b > 0, a, α, β (α < β) ´?¿½~ê. -
C = y(x) ∈ C[α, β]; |y(x)− a| ≤ b,
ρ ´ C þål, =
ρ(x, y) = maxt∈[α,β]
|x(t)− y(t)|, ∀x(t), y(t) ∈ C.
K (C, ρ) ´ålm.
3. y²íØ 12.
51
§2.5 SK
4. Ω ⊂ Rn ´k.4«, b > 0. -
C = y(x) ∈ C(Ω); |y(x)| ≤ b.
3 C þ½Â
ρ(y1, y2) = maxx∈Ω|y1(x)− y2(x)|, ∀ y1, y2 ∈ C.
K (C, ρ) ´ålm.
5. y²·K 15.
6. y²íØ 17.
7. y²½n 18 N5 2).
8. |^½n 18 N5 3) ¥ Picard Sy²½n 18.
9. |^Ø NnÚ½n 18 y²ÏL·Ð©¼êEЯK
y′(x) = y2 + x2, y(0) = 1,
3 Cq) (|^Ø Nn¥SECq)).
10. y² Arzela–Ascoli ÚnN5 1) Ú 2).
11. |^ Arzela–Ascoli ÚnÚ½n 19 N5¥ Tonelli Sy²½n 19.
12. b¼ê f(x, y), F (x, y) 3« Ω ⊂ R2 þëY,
f(x, y) < F (x, y), (x, y) ∈ Ω.
e φ(x), Φ(x) ©O´§
y′ = f(x, y), y′ = F (x, y),
LÓ: (x0, y0) ∈ Ω ), K3§Ó3«mþk
φ(x) < Φ(x), x > x0; φ(x) > Φ(x), x < x0.
52
1Ù ©§)35Ú5
13. δ ∈ Q, ?ØЯK
dy
dx=
0, y = 0,
y lnδ |y|, y 6= 0,y(0) = 0,
δ > 0 Ûk).
14. ?Øe§¤k)3«mµ
14.1. y′ = y2;
14.2. y′ = x2 + y2;
14.3. y′ = 1/(x2 + y2)
53
1nÙ p©§Ú§|)nØ
1Ù?Ø~©§)3Ú5, ±9)'uÐÚëêëY6
5. Ù=©§|Úp©§.
§3.1 p©§Ú§|: )5
Ĺëê n ©§Ð¯K
y(n)(x) = f(x, y, y′, . . . , y(n−1)(x), λ
), (3.1.1)
y(x0) = y0, y′(x0) = y1, . . . , y
(n−1)(x0) = yn−1, (3.1.2)
Ù¥ λ ´ m ëê, x0, y0, y1, . . . , yn−1 ´¢~ê. C
z1 = y(x), z2 = y′(x), . . . , zn = y(n−1)(x).
-
z =
z1
z2
...
zn−1
zn
, z0 =
y0
y1
...
yn−2
yn−1
, f =
z2
z3
...
zn
f(x, z, λ)
.
K n ©§Ð¯K (3.1.1) =z¤ n ©§|ЯK
z′(x) = f(x, z, λ), (3.1.3)
z(x0) = z0. (3.1.4)
l§ (3.1.1) Ú (3.1.3) m'X, N´Xe(Ø.
·K21. éup©§ (3.1.1) Ú©§| (3.1.3) ), e(ؤáµ
a) y = φ(x) ´p§ (3.1.1) ÷vЩ^ (3.1.2) ½Â3,«m J þ)=
z(x) = (φ(x), φ′(x), . . . , φ(n−1)(x))T ,
´§| (3.1.3) ÷vЩ^ (3.1.4) ½Â3 J þ), Ù¥ T L«Ý=.
54
§3.1 p©§Ú§|: )5
b) y1 = φ1(x), . . . , yn = φn(x) ´p§ (3.1.1) 3,«m J þ5Ã')=
z1(x) = (φ1(x), φ′1(x), . . . , φ(n−1)1 (x))T , . . . , zn(x) = (φn(x), φ′n(x), . . . , φ
(n−1)n (x))T ´
§| (3.1.3) 3 J þ5Ã').
d·K 21, e¡ØüÕ?Øp§Ð¯K)nØ, ?ا|)nØ.
aquXþ§, §|ЯKЩ^±=z§|ëê. Ø5,
Ĺëê n ©§|ЯK
y′ = f(x,y, λ), y(0) = 0, (3.1.5)
Ù¥ (x,y) ∈ Ω ⊂ R1+n, λ ∈ Λ ⊂ Rm, Ω, Λ Ñ´m«.
PÒüå, e¡^ fy L« n þ¼ê f 'u n Cþ y Jacobi Ý.
aq/·^PÒ fλ.
½n22. f(x,y, λ), fy ∈ C(Ω × Λ). KЯK (3.1.5) k), P y = φ(x, λ),
(x, λ) ∈ J × Λ, Ù¥ J ´)3«m. ?Ú/, φ(x, λ) 'uÙCþëY.
y: y²aquXþ§, öSÖögC¤.
þã(Ø)û©§|ЯK)'ugCþ!ëêÚЩ^ëY65.
e¡?Ú?ØЯK (3.1.5) )'uÙCþëY5.
½n23. f(x,y, λ), fy, fλ ∈ C(Ω×Λ). KЯK (3.1.5) ), P y = φ(x, λ), '
uÙCþëY.
y: duЯK (3.1.5) ) φ(x, λ) ÷vÈ©§
φ(x, λ) =
∫ x
0
f(s, φ(s, λ), λ)ds, (3.1.6)
φ(x, λ) 'uÙCþëY, ¤± φx(x, λ) 'uÙCþëY.
du´ÛÜ5, ¤±Iy² φ(x, λ)3 Λ¥?: λ0 ,SëY.
duÄ φ 'uÙCþ λ êëY5, PÒüå, Øb λ ´üCþ. P
∆λ0 = λ− λ0. lÈ©§ (3.1.6)
φ(x, λ)− φ(x, λ0)
∆λ0=
∫ x
0
f(s, φ(s, λ), λ)− f(s, φ(s, λ0), λ0)
∆λ0ds (3.1.7)
=
∫ x
0
[(fy[s] + r1(s, λ, λ0))
φ(s, λ)− φ(s, λ0)
∆λ0+ fλ[s] + r2(s, λ, λ0)
]ds,
55
1nÙ p©§Ú§|)nØ
Ù¥ [s] = (s, φ(s, λ0), λ0), r1 Ú r2 'uÙCþëY,
lim∆λ0→0
r1(s, λ, λ0) = 0, lim∆λ0→0
r2(s, λ, λ0) = 0.
P
z(x, λ, λ0) =φ(x, λ)− φ(x, λ0)
∆λ0.
Kk
dz(x, λ, λ0)
dx= (fy[x] + r1(x, λ, λ0)) z(x, λ, λ0) + fλ[x] + r2(x, λ, λ0), z(0, λ, λ0) = 0. (3.1.8)
Ï fy[x], fλ[x] 'uÙCþëY, ¤±d½n 22 , ЯK (3.1.8) ) z(x, λ, λ0) '
uÙCþëY. 4
lim∆λ0→0
φ(x, λ)− φ(x, λ0)
∆λ0= limλ→λ0
z(x, λ, λ0)∃,
= ê φλ(x, λ0) 3. d λ0 À?¿5, ê φλ(x, λ), λ ∈ Λ, 3.
é§ (3.1.7) 4 λ→ λ0 ,
φλ(x, λ0) =
∫ x
0
fy(s, φ(s, λ0), λ0)φλ(s, λ0) + fλ(s, φ(s, λ0), λ0)ds. (3.1.9)
Ïd^ λ O λ0, È©§ (3.1.9) duЯK
dφλ(x, λ)
dx= fy(x, φ(x, λ), λ)φλ(x, λ) + fλ(x, φ(x, λ), λ), φλ(0, λ) = 0. (3.1.10)
d φ(x, λ) ëY5Ú½n 22 φλ(x, λ) 'uÙCþëY. y..
aqu½n 23 y²Xe(Ø
íØ24. éuw¹Ð©^ЯK
y′ = f(x,y, λ), y(x0) = y0. (3.1.11)
b f(x,y, λ), fy ∈ C(Ω× Λ), y = φ(x, λ, x0,y0) ´ (3.1.11) ). K
(a) φ 'u x0 ê φx0´Ð¯K
dz
dx= fy(x, φ(x, λ, x0,y0), λ)z, z(x0) = −f(x0,y0, λ). (3.1.12)
). Ï φ(x, λ, x0,y0) 'u x0 këY ê.
56
§3.2 )Û©§)nØ
(b) φ 'u y0 ê (O(/` Jacobi Ý) φy0´ÝЯK
dZ
dx= fy(x, φ(x, λ, x0,y0), λ)Z, Z(x0) = E. (3.1.13)
). Ï φ(x, λ, x0,y0) 'u y0 këY ê.
yµöS, ÖögC¤. J«: ЯK (3.1.12) Ú (3.1.13) ¥Ð©^dé
φ(x, λ, x0,y0)− y0 =
∫ x
x0
f(s, φ(s, λ, x0,y0), λ)ds,
'u x0 Ú y0 ©O¦ ê, 2- x = x0 .
N5: 5©§ (3.1.10), (3.1.12) Ú (3.1.13) ©O¡§ (3.1.5) 'uëê λ, Щ
x0 Ú y0 C©§. C©§3©§)5ïÄ¡åX~^.
~K: b f(x, y) ∈ C1(Ω), Ω ⊂ R2 ´m«. e φ(x, x0, y0), x ∈ J ´Ð¯K
y′ = f(x, y), y(x0) = y0,
), K
∂φ(x, x0, y0)
∂x0+ f(x0, y0)
∂φ(x, x0, y0)
∂y0≡ 0, x ∈ J.
yµdíØ 24 ,∂φ(x, x0, y0)
∂x0Ú
∂φ(x, x0, y0)
∂y0©O÷vЯK
dz
dx= fy(x, φ(x, x0, y0))z, z(x0) = −f(x0, y0),
Ú
dz
dx= fy(x, φ(x, x0, y0))z, z(x0) = 1.
¦)ü5©§Ð¯K
∂φ(x, x0, y0)
∂x0= −f(x0, y0)e
∫ xx0fy(s,φ(s,x0,y0))ds
,
∂φ(x, x0, y0)
∂y0= e
∫ xx0fy(s,φ(s,x0,y0))ds
.
|^ùüª=(Ø. y..
½n 23 )û~©§)'uÙgCþ!ëê9Щ^ëY5.
, ·ke¡(Ø, Ùy²lÑ.
½n25. f(x,y, λ) ∈ Ck(Ω× Λ), k ∈ N ∪ ∞. KЯK (3.1.5) )'ugCþ x Ú
ëê λ Ñ´Ck ëY.
57
1nÙ p©§Ú§|)nØ
§3.2 )Û©§)Û)
þ!?Ø1w©§1w)35. !?Ø)Û©§)Û)35.
éu n þ y Ú y0, e¡ò©O^ yi Ú yi0 L«§1 i ©þ.
¼ê f(x,y)3« Ω ⊂ R1+nS)Û,XJé ∀ (x0,y0) ∈ Ω, ∃α > 0, β > 0¦ f(x,y)
3
D := (x,y); |x− x0| ≤ α, |yi − yi0| ≤ β, i = 1, . . . , n ⊂ Ω,
S±Ðm¤ x− x0, y − y0 Âñ?ê
f(x,y) =
∞∑i,|j|=0
aij(x− x0)i(y − y0)j,
Ù¥ (y − y0)j = (y1 − y10)j1 · · · (yn − yn0)jn , |j| = j1 + . . .+ jn.
½?ê
∞∑i,|j|=0
aij(x− x0)i(y − y0)j, (3.2.1)
∞∑i,|j|=0
Aij(x− x0)i(y − y0)j, (3.2.2)
• XJ |aij| ≤ Aij, ¡ (3.2.2) ´ (3.2.1) `?ê;
• XJ (3.2.2)´ (3.2.1)`?ê, (3.2.2)3 DSÂñ,PÚ¼ê F (x,y),¡ F (x,y)
´ (3.2.1) 3 D S`¼ê.
e¡Ä)Û©§Ð¯K)Û)35. duЩ^o±ÏLC=z
§¥, Ø5·Ä)Û©§lI:Ñu).
½n26. fi(x,y), i = 1, . . . , n, 3 D SФÂñ?ê. KЯK
yi = fi(x,y), yi(0) = 0, i = 1, . . . , n, (3.2.3)
3 (0,0) ,SkÂñ?ê), Ù¥ 0 L« n "þ.
y: 1. y²é?¿ 0 < a < α, 0 < b < β, 73 M > 0 ¦
G(x,y) =M(
1− xa
) (1− y1
b
). . .(1− yn
b
) ,58
§3.2 )Û©§)nØ
´ fi(x,y), i = 1, . . . , n, 3
D0 = (x,y); |x| ≤ a, |yi| ≤ b, i = 1, . . . , n ⊂ D,
þ`¼ê. ¯¢þ, db
fi(x,y) =
∞∑k0+|k|=0
a(i)k0k
xk0yk, i = 1, . . . , n, (3.2.4)
3 D þÂñ. ¤±∞∑
k0+|k|=0
∣∣∣a(i)k0k
∣∣∣ ak0bk1 . . . bkn , i = 1, . . . , n,
Âñ. - M ù n þ. Ké¤k k0,k Ú i Ñk
∣∣∣a(i)k0k
∣∣∣ ≤ M
ak0bk1 . . . bkn.
l (x,y) ∈ D0 \ ∂D0
G(x,y) =
∞∑k0+|k|=0
M
ak0bk1 . . . bknxk0yk =
M(1− x
a
) (1− y1
b
). . .(1− yn
b
) ,´ fi(x,y), i = 1, . . . , n, `¼ê.
2. ´
dz
dx=
M(1− x
a
) (1− z
b
)n , z(0) = 0,
3«m |x| < ρ := a(
1− e−b
(n+1)aM
)Sk)Û)
z(x) = b
(1−
(abM(n+ 1) ln
(1− x
a
)+ 1) 1n+1
).
l d)5½n, y = (y1, . . . , yn) = (z(x), . . . , z(x)) ´Ð¯K
y′i(x) = G(x,y), yi(0) = 0, i = 1, . . . , n, (3.2.5)
3 |x| < ρ S)Û).
3. y (3.2.3) 3 |x| < ρ Sk/ª?ê).
yi(x) =
∞∑j=0
c(i)j xj , i = 1, . . . , n, (3.2.6)
59
1nÙ p©§Ú§|)nØ
´Ð¯K (3.2.3) /ª). K
c(i)j =
1
j!
dj yidxj
∣∣∣∣x=0
, j = 0, 1, . . . ,
ò/ª) (3.2.6)\ЯK (3.2.3), |^ fi Ðmª (3.2.4), ¿é§ (3.2.3)ü>'u
x Åg¦ê
c(i)0 = yi(0) = 0,
c(i)1 =
dyidx
∣∣∣∣x=0
= fi(x, y)|x=0 = a(i)000...0,
c(i)2 =
1
2!
d2yidx2
∣∣∣∣x=0
=1
2!
(∂fi∂x
+∂fi∂y1
dy1
dx+ . . .+
∂fi∂yn
dyndx
)∣∣∣∣x=0
=1
2!
(a
(i)100...0 + a
(i)010...0a
(1)000...0 + . . .+ a
(i)000...1a
(n)000...0
),
c(i)j =
1
j!
dj yidxj
∣∣∣∣x=0
= P(i)j
(a
(1)00...0, . . . , a
(n)00...0, a
(1)10...0, . . . , a
(n)10...0, . . . , a
(n)00...j−1
),
Ù¥ j ≥ 2, P(i)j ´± f1, . . . , fn ÐmªXê a
(s)k0k
, k0 + |k| ≤ j − 1, s ∈ 1, . . . , n, Cþ
(½Xêõª.
4. y (3.2.3) 3 |x| < ρ S/ª?ê)Âñ.
G(x,y) 3 |x| < ρ S?êÐmª
G(x,y) =
∞∑k0+|k|=0
gk0kxk0yk.
ЯK (3.2.5) 3 |x| < ρ S)Û)?êÐmª
yi(x) =
∞∑j=0
c(i)j xj , i = 1, . . . , n.
Kaqu (3.2.3) /ª)¦
c(i)j = P
(i)j (g00...0, . . . , g00...0, g10...0, . . . , g10...0, . . . , g00...j−1) , i = 1, . . . , n, j = 0, 1, . . .
du c(i)j Ú c
(i)j ´Xêõª P
(i)j 3ØÓ:, ∣∣∣a(i)
k0k
∣∣∣ ≤ gk0k =M
ak0bk1 . . . bkn, k = (k1, . . . , kn),
¤± |c(i)j | ≤ c(i)j . ùÒy²Ð¯K (3.2.5) )Û) yi(x), i = 1, . . . , n, ´Ð¯K
(3.2.3) /ª) yi(x), i = 1, . . . , n, 3 |x| < ρ S`¼ê. (3.2.3) /ª?ê)
(3.2.6) 3 |x| < ρ SÂñ, l ЯK (3.2.3) 3 |x| < ρ Sk)Û). y..
60
§3.2 )Û©§)nØ
N5: þãy²/©z [16] é)Û©§)Û)35y².
~K:
1. ©§|
dy1
dx= ex
2+y21+y21 + 5 cos(y1y2),dy2
dx= lnx+ ln(y2
1 + y22),
l (0,∞)×(R2 \ (0, 0)
)¥?: (x0, y
01 , y
02) ÑuÑk½Â3 x0 ,S
)Û). ù´Ï§|¥¼êѱ3 (x0, y01 , y
02) ,SФÂñ?
ê, l d½n 26 =y.
e(JéaAÏ)Û©§|Ø=Ñ)3«m Ñ?ê)Â
ñ».
½n27. b A(x) Ú f(x) ©O´ n ݼêÚ n þ¼ê, §z©þ3
|x− x0| < ρ SÑФ'u x− x0 Âñ?ê, K©§|ЯK
dy
dx= A(x)y + f(x), y(x0) = y0, (3.2.7)
3 |x− x0| < ρ SkÂñ?ê).
y: PÒüå, Ø5, ·b x0 = 0,y0 = 0. P
A(x) = (aij(x))n×n, f(x) = (f1(x), . . . , fn(x))T ,
Ù¥ T L«=. db, é i, j ∈ 1, . . . , n k
aij(x) =
∞∑k=0
a(ij)k xk, fj(x) =
∞∑k=0
f(j)k xk, |x| < ρ.
é?¿ b ∈ (0, ρ), e~ê?ê
∞∑k=0
|a(ij)k |b
k,
∞∑k=0
|f (j)k |b
k, i, j ∈ 1, . . . , n,
Âñ. - M §. Ké¤k i, j ∈ 1, . . . , n, k ∈ 0, 1, 2, . . . k
|a(ij)k |, |f
(j)k | ≤
M
bk.
Ï
g(x) =M
1− xb
=
∞∑k=0
M(xb
)k, |x| < b,
61
1nÙ p©§Ú§|)nØ
´ aij(x) Ú fj(x) 3 |x| < b þ`¼ê. ?Ú/k
g(x)(y1 + . . .+ yn + 1),
´n∑j=1
aij(x)yi + fi(x), i = 1, . . . , n,
`¼ê.
O, ЯK
du
dx= g(x)(nu+ 1), u(0) = 0,
k)
u(x) = n−1(
1− x
b
)−nMb
− n−1.
dêÆ©Û£, u(x) 3 |x| < b þФÂñ?ê.
d)5nØ, y∗(x) = (u(x), . . . , u(x)) ´©§|ЯK
dyidx
= g(x)(y1 + . . .+ yn + 1), yi(0) = 0, i = 1, . . . , n,
).
-
y(x) = (y1(x), . . . , yn(x))T , yi(x) =
∞∑k=0
c(i)k xk, i = 1, . . . , n,
´Ð¯K (3.2.7) /ª). K½n 26 Óy², u(x) ´ yi(x), i = 1, . . . , n, 3
|x| < b þ`¼ê. Ï yi(x), i = 1, . . . , n, ´ |x| < b þÂñ?ê. d)5,
y(x) ´Ð¯K (3.2.7) 3 |x| < b þÂñ?ê). d b ∈ (0, ρ) ?¿5, ЯK
(3.2.7) 3 |x| < ρ þkÂñ?ê). y..
§3.3 ©§ÈnØ
©§ÈnØQ´²;, q´y. §éXyêÆ©|, X²;
Úy©Û, ©AÛÚ Riemann AÛ, êÿÀÚêAÛ. k,Ööë
[5–7, 9, 12, 18, 38, 47, 52] 9Ù¥ë©z. !0Ù¥ÄVgÚnØ.
62
§3.3 ©§ÈnØ
Ä n ©§|
dyidx
= fi(x,y), (x,y) ∈ D, i = 1, . . . , n, (3.3.1)
Ù¥ D ⊂ R1+n´m«.y)35,!©ªb fi ∈ C1(D), i = 1, . . . , n.
G ´ D f«. ½Â3 G þ¼ê V (x,y) ¡§ (3.3.1) ÄgÈ©, XJ
• V (x,y) ∈ C(G), 3 G ?¿f«¥ÑØð~ê,
• § (3.3.1) 3 G ¥?Û^È© Γ := φ(x) = (y1(x), . . . , yn(x)); x ∈ J Ñk
V (x, φ(x)) ≡ CΓ, x ∈ J , Ù¥ CΓ ´6u Γ ~ê.
N5:
1) XJ3ÄgÈ©½Â¥, V (x,y) 3 G ,f« Ω ¥ðu~ê, Kl V Ø
§ (3.3.1) 3 Ω ¥?Û&E.
2) ¦+§| (3.3.1) ½Â3 D þ, ÙÄgÈ©7U½Â3 D þ (e¡~
2). ù´3 D f«þ½Â (3.3.1) ÄgÈ©Ï.
3) XJ V (x,y) ´ (3.3.1) 3 G þÄgÈ©, h(z) ´?ëY¼ê¦ h(V (x,y))
3 G ?¿f«þØðu~ê, K h(V (x,y)) ´ (3.3.1) 3 G þÄgÈ
©.
e¡(ØÑëYÄgÈ©d½.
·K28. b V 3 G ¥ëY, 3 G ?Ûf«þØðu~ê. K V ´§
(3.3.1) 3 G ¥ÄgÈ©¿^´
∂V (x,y)
∂x+∂V (x,y)
∂y1f1(x,y) + . . .+
∂V (x,y)
∂ynfn(x,y) ≡ 0, (x,y) ∈ G.
y: öSÖögC¤.
~K:
1. ©§|
dx
dt= −y, dy
dt= x,
3 R2 ¥kÄgÈ© V (t, x, y) = x2 + y2. Ù?Û^È©Ñd, V (t, x, y) = c Ñ,
l ½ö´ x = 0, y = 0, ½ö´±:¥%±.
63
1nÙ p©§Ú§|)nØ
2. ¦©§|
dx
dt= y − x(x2 + y2 − 1)(x2 + y2 − 2),
dy
dt= −x− y(x2 + y2 − 1)(x2 + y2 − 2), (3.3.2)
ÄgÈ©, ¿^ÄgÈ©½§| (3.3.2) È©.
): l§| (3.3.2) Lª
xx′(t) + yy′(t) = −(x2 + y2)(x2 + y2 − 1)(x2 + y2 − 2), yx′(t)− xy′(t) = x2 + y2.
éþªÈ© (3.3.2) ÄgÈ©
V1(t, x, y) = (x2 + y2)(x2 + y2 − 2)(x2 + y2 − 1)−2e4t, V2(t, x, y) = arctany
x+ t.
5¿: ¦+§| (3.3.2)3 R3 þk½Â)Û,ÙÄgÈ©3 R3 f«þk½Â.
|^4IC x = r cos θ, y = r sin θ, l V1 = c1 Ú V2 = c2 )
x =
√1± 1√
1− c1e−4tcos(c2 − t), y =
√1± 1√
1− c1e−4tsin(c2 − t). (3.3.3)
¤±§| (3.3.2) È©©ÙXeµ
• c1 = 0, (3.3.3) éA (3.3.2) ~ê)µx = 0, y = 0 Ú È©µ± Γ2 := x =√
2 cos(c2 − t), y =√
2 sin(c2 − t).
• c1 > 0, (3.3.3) éA (3.3.2) ^È©, T t→∞ ^%C Γ2.
• c1 < 0, (3.3.3) éA (3.3.2) ü^È©. Ù¥^ uü ± Γ1 Ú Γ2 m,
t → ∞ ^%C Γ2, t → −∞ ^%C Γ1. ,^ uI:Ú Γ1
m, t→∞ ^%CI:, t→ −∞ ^%C Γ1.
• c1 = −∞, (3.3.3) éA (3.3.2) È© Γ1.
N5:
1) ~ 2¥È© Γ1 Ú Γ2 Ñ´§| (3.3.2)±Ï),3ùü±Ï)
SvkÙ§±Ï). ù±Ï)¡4.
2) þã~fÖöùéuµ½§|võÄgÈ©,Ò±|^
§é§¦). e¡ò?ØÄgÈ©35.
64
§3.3 ©§ÈnØ
3) p©§
y(n)(x) = f(x, y, y′, . . . , y(n−1)
),
±ÏLC y1 = y, y2 = y′, . . . , yn = y(n−1) =z§|, Ïd§ÄgÈ© (b
3) /ª V (x, y, y′, . . . , y(n−1)).
§3.3.1 ÈÄ:nصÄgÈ©35
V1(x,y), . . . , Vk(x,y)´§| (3.3.1)3 GSëYÄgÈ©. XJ V1, . . . , Vk
'u y Jacobi Ý´÷, =
rank∂(V1, . . . , Vk)
∂(y1, . . . , yn)= k,
¡ V1, . . . , Vk ´§| (3.3.1) 3 G S k ¼êÕáÄgÈ©.
N5: ¼êÕáÄgÈ©AÛ)º.
1) V (x,y)´§| (3.3.1)3 GSÄgÈ©. XJ Sc := (x,y) ∈ G; V (x,y) =
c ´¡, ¡ V ³¡. üå, ±^ V = c L«³
¡ Sc. ´l V ,³¡þÑuÈ©©ª±3T³¡þ. äkù5
¡¡§ (3.3.1) ØC¡.
2) V1(x,y), . . . , Vk(x,y) ´§| (3.3.1) 3 G SëY¼êÕáÄgÈ©, K
§³¡3:?Ñ´î, =¡Ø. ù³¡8´ n− k
¡.
½n29. fi(x,y) ∈ C1(D), i = 1, . . . , n. Ké ∀P0 = (x0,y0) ∈ D, 3 P0
G0 ⊂ D ¦ (3.3.1) 3 G0 Sk=k n ¼êÕáÄgÈ©.
y: 1. y²§| (3.3.1) 3 P0 ,k n ¼êÕáÄgÈ©.
é P0 ? D0 ⊂ D, ? (x0, c) ∈ D0. §| (3.3.1) ÷vЩ^
y(x0) = c,
)3 D0 ¥3'uÙCþëY, P y = φ(x, c), x ∈ J .
65
1nÙ p©§Ú§|)nØ
Ï φ = (φ1, . . . , φn),
φi(x, c) = φi(x0, c) + ∂xφi(x0 + θ(x− x0), c)(x− x0)
= ci + ∂xφi(x0 + θ(x− x0), c)(x− x0),
¤± φ 'u c Jacobi 1ª
∂φ(x, c)
∂c
∣∣∣∣x=x0
= E,
Ù¥ E ´ü Ý. q
φ(x0,y0) = y0,
¤±dÛ¼ê3½n, 3 P0 G0 ⊂ D0 ¦3Ù¥¼ê§ φ(x, c) = y k
ëY)
c = V(x,y).
ey V ¥ n ¼ê´¼êÕáÄgÈ©. Ï
φ(x,V(x,y)) ≡ y, (x,y) ∈ G0, (3.3.4)
¤±éT¼ê§ü>'u y ¦ê
∂V
∂y=
(∂φ
∂c
)−1
.
ùÒy² V ¥ n ¼ê´¼êÕá. é¼ê§ (3.3.4) ü>'u x ¦ê
∂φ
∂c
∂V1
∂x
...
∂Vn∂x
=
−∂φ1
∂x
...
−∂φn∂x
=
−f1
...
−fn
,
¤± ∂V1
∂x
...
∂Vn∂x
=
(∂φ
∂c
)−1
−f1
...
−fn
=∂V
∂y
−f1
...
−fn
.
ùÒy² Vi(x,y), i = 1, . . . , n, ´ (3.3.1) n ÄgÈ©.
2. y§| (3.3.1) õk n ¼êÕáÄgÈ©.
66
§3.3 ©§ÈnØ
bk n+ 1 ÄgÈ© Vi(x,y), i = 1, . . . , n+ 1. du§'u y Jacobi Ý
∂V1
∂y1. . . ∂V1
∂yn...
. . ....
∂Vn∂y1
. . . ∂Vn∂yn
∂Vn+1
∂y1. . . ∂Vn+1
∂yn
,
≤ n, ¤± V1, . . . , Vn+1 3?Û«¥ÑØU¼êÕá.
½ny..
N5:
1) XJ f(x,y) = (f1(x,y), . . . , fn(x,y)) 6≡ 0, § (3.3.1) عgCþ¼êÕáÄg
È©õk n− 1 .
2) ½n 29 ѧ| (3.3.1) 3½Â D ¥,:¥ÄgÈ©ÛÜ35.
3 D ¥, ½3 D ,½«¥ÄgÈ©N35´~(J¯K.
3) ½n 29 y²þ´|^§| (3.3.1) )ÏLÛ¼ê3½nE n ¼ê
ÕáÄgÈ©. éA, ·ke¡(Ø.
½n30. Vi(x,y), i = 1, . . . , n, ´§| (3.3.1) 3 G þëY¼êÕáÄgÈ
©. KdÛ¼ê3½nl Vi(x,y) = ci, i = 1, . . . , n, )Ѽê
y = φ(x, c), x ∈ Jc, Ù¥ c ´?¿~êþ, (3.3.5)
´ (3.3.1) 3 G SÏ), ¹ (3.3.1) 3 G S¤k).
y: 1. y (3.3.5) ´§| (3.3.1) Ï).
Ï Vi (i = 1, . . . , n) ëY, ¤± φ = (φ1, . . . , φn) 'uÙCþëY. é¼ê
§
Vi(x, φ(x, c)) ≡ ci, i = 1, . . . , n,
ü>'u x ¦ê
∂Vi∂x
+∂Vi∂y1
∂φ1
∂x+ . . .+
∂Vi∂yn
∂φn∂x≡ 0, i = 1, . . . , n.
q Vi ´ëYÄgÈ©, ¤±k
∂Vi∂x
+∂Vi∂y1
f1 + . . .+∂Vi∂yn
fn ≡ 0, i = 1, . . . , n.
67
1nÙ p©§Ú§|)nØ
lþãü¼ê§|
∂V
∂y
∂φ1
∂x − f1
...
∂φn∂x − fn
= 0.
du V1, . . . , Vn ´¼êÕá, þãàg5§|k)
dφidx− fi = 0, i = 1, . . . , n,
ùÒy² (3.3.5) ´ (3.3.1) ). (3.3.5) ´ (3.3.1) Ï)d
det∂φ
∂c=
(det
∂V
∂y
)−1
6= 0,
.
2. y (3.3.5) ¹ (3.3.1) ¤k).
y = ψ(x) ´ (3.3.1) ). - y0 = ψ(x0), K y = ψ(x) ´ (3.3.1) ÷vЩ^
y(x0) = y0 ).
- c0 = V(x0,y0). KdÛ¼ê3½nl¼ê§
V(x,y) = c0,
) y = φ(x, c0) ´©§| (3.3.1) ), ÷v φ(x0, c0) = y0. d)5
φ(x, c0) = ψ(x). ½ny..
N5:
½Â3« D þ©§| (3.3.1) ÄgÈ©73 D þk½Â. Ïd3
ÄgȩýÂ8Ü¥§| (3.3.1) )I,?Ø. X~ 2 ¥ÄgÈ© V1 3
x2 + y2 = 1 þýÂ, ÄgÈ© 1/V1 3 x2 + y2 = 1 þk½Â, %3 x2 + y2 = 0 Ú
x2 + y2 = 2 þýÂ.
½n 29)û n©§|ÛÜÄgÈ©ê¯K.e¡½n?Úx¼
êÕáÄgÈ©.
½n31. V1(x,y), . . . , Vn(x,y)´ (3.3.1)3 G¥ëY¼êÕáÄgÈ©, Φ(x,y)
´ (3.3.1) 3 G ¥?ëYÄgÈ©. Ké ∀P0 = (x0,y0) ∈ G, 3 P
G0 ⊂ G, 9ëY¼ê h(z) ¦ Φ(x,y) = h(V1(x,y), . . . , Vn(x,y)), (x,y) ∈ G0.
68
§3.3 ©§ÈnØ
y: du V1(x,y), . . . , Vn(x,y) 3 G ¥ëY¼êÕá, ¤±¼ê§
Vi(x,y) = ui, i = 1, . . . , n, Ù¥ u = (u1, . . . , un) ÕáCþ,
3 P0 ,, P G0, ¥këY), P
y = φ(x,u), x ∈ J, u ∈ Ω := V(x,y); (x,y) ∈ G0.
d½n 30 , φ(x,u) ´§ (3.3.1) 3 G S), =
∂φi(x,u)
∂x= fi(x, φ(x,u)), (x,u) ∈ J × Ω, i = 1, . . . , n,
Ù¥ u ´ëê.
- h(x,u) = Φ(x, φ(x,u)), (x,u) ∈ J × Ω, ¿é h 'u x ¦ ê
∂h(x,u)
∂x=
∂Φ[x,u]
∂x+∂Φ[x,u]
∂y1
∂φ1(x,u)
∂x+ . . .+
∂Φ[x,u]
∂yn
∂φn(x,u)
∂x
=∂Φ[x,u]
∂x+∂Φ[x,u]
∂y1f1[x,u] + . . .+
∂Φ[x,u]
∂ynfn[x,u] ≡ 0, (x,u) ∈ J × Ω,
Ù¥ðªd [x,u] = (x, φ(x,u)) ∈ G0 ⊂ G Ú Φ(x,y) ´ (3.3.1) 3 G ¥Äg
È©. ùÒy² (x,y) ∈ G0 k
Φ(x,y) = Φ(x, φ(x,u)) = h(u) = h(V1(x,y), . . . , Vn(x,y)).
½ny..
N5:
1) lþã½n(Ø, §| (3.3.1) 3 G ⊂ R1+n ¥k n ¼êÕáÄgÈ©,
K§È©dùÄgÈ©³¡(½. d¡§| (3.3.1) 3 G ¥
È.
2) XJÄÏCþ y ¤3m Ξ ⊂ Rn (¡m, AO/ n = 2 ¡
²¡), §| (3.3.1) 3 Ξ ¥k n− 1 عgCþ x ¼êÕáÄgÈ©, K
m Ξ ¥È© (¡;) dù n− 1 ¼êÕáÄgÈ©(½. d
¡§| (3.3.1) 3 Ξ ¥È.
3) ¦+3 1) Ú 2) ¥ÑÈü½Â, 7L`²8cÈvk
Ú½Â,lØÓÝÑukØÓ½Â.e!?Ø HamiltonXÚ,é
ÈÑ#½Â.
69
1nÙ p©§Ú§|)nØ
4) §| (3.3.1) 3 G ¥k 1 ≤ k < n ëY¼êÕáÄgÈ©, K3
ù k ÄgÈ©³¡(½ n − k ØC¡þ, §| (3.3.1) ±z¤
n − k ©§|. X3~ 1 ÄgÈ©(½³ x2 + y2 = c > 0, =
y =√c− x2 þ, §|z©§
x′(t) = −√c− x2, x ∈ (−
√c,√c).
¯¢þ, 3þã³þ1§±d1§, Ï
y′(t) =−xx′(t)√c− x2
= x.
5) 'ug£©XÚÄgÈ©m'X, ©z [38] k?ÚïÄ.
6) ©z [21]éa²;¿ÂeÈk¡0,Ï´Ã.íéÈk,
ÖöÖ.
§3.3.2 ÄgÈ©3 ©§¦)¥A^
ÈnØ3 ©§¦)¥A^, !Ì?Øàg5 ©§
n∑i=1
ai(x)∂u
∂xi= 0, x = (x1, . . . , xn) ∈ D ⊂ Rn m«, (3.3.6)
Ú[5 ©§
n∑i=1
ai(x, u)∂u
∂xi= b(x, u), (x, u) ∈ G ⊂ Rn+1 m«, (3.3.7)
¦)¯K.
§3.3.2.1 àg5 ©§ (3.3.6) )nØ
©§ (3.3.6) éAA§´
dx1
a1= . . . =
dxnan
. (3.3.8)
b
a1, . . . , an ∈ C1(D), n∑i=1
|ai(x)| > 0, x ∈ D. (3.3.9)
70
§3.3 ©§ÈnØ
K (3.3.8) ´ n− 1 ~©§|. ~X an(x) 6= 0 , (3.3.8) ±¤
dxidxn
=ai(x)
an(x), i = 1, . . . , n− 1.
Ïd§| (3.3.8) l D ¥?:Ñu)Ñ3. ?Ú/, (3.3.8) ÛÜ/k n − 1
¼êÕáÄgÈ©.
½n32. b (3.3.8) ÷v (3.3.9), 3 D ¥k n− 1 ¼êÕáÄgÈ©
φ1(x) = c1, . . . , φn−1(x) = cn−1.
K5 ©§ (3.3.6) Ï)
u = Ψ(φ1(x), . . . , φn−1(x)),
Ù¥ Ψ(·, . . . , ·) ´?¿ n− 1 ëY¼ê.
y: d®^, Ø an 6= 0. Ï A§ (3.3.8) du~©§|
dxidxn
=ai(x)
an(x), i = 1, . . . , n− 1. (3.3.10)
âëYÄgÈ©d½, φ(x) ´ (3.3.10) ÄgÈ©= φ(x) ´
©§
∂φ
∂xn+a1(x)
an(x)
∂φ
∂x1+ . . .+
an−1(x)
an(x)
∂φ
∂xn−1= 0,
i.e.
a1(x)∂φ
∂x1+ . . .+ an−1(x)
∂φ
∂xn−1+ an(x)
∂φ
∂xn= 0,
). Ïd¦ ©§ (3.3.6) Ï)du¦~©§| (3.3.10) ¤kÄgÈ©.
e φ(x) ´ (3.3.10) ÄgÈ©, Kd½n 31 , 3ëY n− 1 ¼ê Ψ ¦
φ(x) = Ψ(φ1(x), . . . , φn−1(x)).
ùÒy² (3.3.6) Ï)´'u φ1(x), . . . , φn−1(x) ?¿ëY¼ê. ½ny..
N5µ
• ½n 32 ¥Ï)Lª´ÛÜ.
71
1nÙ p©§Ú§|)nØ
• 5 ©§Ï)dÙA§ n − 1 ¼êÕáÄgÈ©?¿
ëY¼ê5L«.
~K:
1. ¦e ©§Ï)µ
x∂u
∂x+ y
∂u
∂y+ (z −
√x2 + y2 + z2)
∂u
∂z= 0. (3.3.11)
)µ ©§ (3.3.11) A§
dx
x=dy
y=
dz
z −√x2 + y2 + z2
.
l
dx
x=dy
y,
A§ÄgÈ© φ1(x, y, z) = xy . òA§C/
xdx
x2=ydy
y2=
(z +√x2 + y2 + z2)dz
−(x2 + y2).
l k
xdx+ ydy + (z +√x2 + y2 + z2)dz = 0, = d(x2 + y2 + z2) + 2
√x2 + y2 + z2dz = 0.
ùA§1ÄgÈ©
φ2(x, y, z) = z +√x2 + y2 + z2.
N´y φ1 φ2 ´¼êÕá. Ï ©§ (3.3.11) Ï)
u(x, y, z) = ψ
(x
y, z +
√x2 + y2 + z2
),
Ù¥ ψ ´?¿ëY¼ê.
2. ¦ ©§
y∂u
∂x+ z
∂u
∂z= 0, (3.3.12)
ÏL¡ x = 1, u = ln z − 1y ).
72
§3.3 ©§ÈnØ
): ©§ (3.3.12) A§
dx
y=dy
0=dz
z.
´ φ1(x, y, z) = y ´A§ÄgÈ©. duA§?^È©Ñ
uÄgÈ©,³¡þ, ¤±3³¡ y = c1 þ, l
dx
y=dz
z,
)
x
c1= ln z + c.
¤±A§,¼êÕáÄgÈ©
φ2(x, y, z) =x
y− ln z.
§Ï)
u = ψ
(y,x
y− ln z
),
Ù¥ ψ ´?¿ëY¼ê.
d®^
ψ
(y,
1
y− ln z
)= ln z − 1
y.
3¼ê ψ ¥- ξ = y, η = 1y − ln z , Kk y = ξ, z = e
1ξ−η. l ¼ê
ψ(ξ, η) = −η.
§÷v½^)
u = −η = ln z − x
y.
§3.3.2.2 [5 ©§ (3.3.7) )nØ
?Ø[5 ©§ (3.3.7), =
n∑i=1
ai(x, u)∂u
∂xi= b(x, u), (x, u) ∈ G ⊂ Rn+1 m«,
73
1nÙ p©§Ú§|)nØ
)35, b
a1, . . . , an, b ∈ C1(G), n∑i=1
|ai(x, u)| > 0, (x, u) ∈ G. (3.3.13)
~©§|
dx1
a1(x, u)= . . . =
dxnan(x, u)
=du
b(x, u), (3.3.14)
¡[5 ©§ (3.3.7) A§.
½n33. b (3.3.14) ÷v (3.3.13), 3 D ¥k n ¼êÕáÄgÈ©
φ1(x, u) = c1, . . . , φn(x, u) = cn.
K[5 ©§ (3.3.7) Ï)
Ψ(φ1(x, u), . . . , φn(x, u)) = 0, (3.3.15)
Ù¥ Ψ(·, . . . , ·) ´?¿ n ëY¼ê.
y: dbÚ½n 31 , A§ (3.3.14) ¤kÄgÈ©
V (x, u) = Ψ(φ1(x, u)), . . . , φn(x, u)),
Ù¥ Ψ ´?¿ëY n ¼ê. l½n 32 9Ùy², V (x, u) ´5 ©§
a1(x, u)∂V
∂x1+ . . .+ an(x, u)
∂V
∂xn+ b(x, u)
∂V
∂u= 0, (3.3.16)
Ï).
e¡Äky²l (3.3.15) âÛ¼ê3½n(½¼ê u = φ(x) ´ (3.3.7) )(
,¦∂V
∂u6= 0). dÛ¼ê§(½¼êê÷v
∂φ(x)
∂xi= − ∂V
∂xi/∂V
∂u, i = 1, . . . , n.
òþãLª\ (3.3.16)
a1(x, φ(x))∂φ
∂x1+ . . .+ an(x, φ(x))
∂φ
∂xn= b(x, φ(x)).
ùÒy² u = φ(x) ´[5 ©§ (3.3.7) ).
74
§3.3 ©§ÈnØ
y² (3.3.7) ?)±L«¤ (3.3.15) /ª. u = ψ(x) ´ (3.3.7) ?
). -
Φi(x) = φi(x, ψ(x)), i = 1, . . . , n.
Ké i = 1, . . . , n k
a1∂Φi∂x1
+ . . .+ an∂Φi∂xn
= a1∂φi∂x1
+ . . .+ an∂φi∂xn
+∂φi∂u
(a1∂ψ
∂x1+ . . .+ an
∂ψ
∂xn
)= a1
∂φi∂x1
+ . . .+ an∂φi∂xn
+ b∂φi∂u
= 0,
Ù¥1ª¥^ ψ(x)´ (3.3.7)),1nª¥^ φi(x, u)´A§ (3.3.14)
ÄgÈ©.
ùÒy² Φ1(x), . . . ,Φn(x) ´ n− 1 ~©§|
dx1
a1= . . . =
dxnan
,
ÄgÈ©, ÏdÙ¥,ÄgÈ©´Ù§ÄgÈ©ëY¼ê. Ø
Φn(x) = Γ(Φ1(x), . . . ,Φn−1(x)),
Ù¥ Γ ´ëY¼ê. -
Ψ(φ1(x, u), . . . , φn(x, u)) = φn(x, u)− Γ(φ1(x, u), . . . , φn−1(x, u)).
K u = ψ(x) ´§
Ψ(φ1(x, u), . . . , φn(x, u)) = 0,
). y..
~Kµ
1. ¦) ©§Ð¯Kµ
√x∂z
∂x+√y∂z
∂y= z,
z|y=1 = cos(ωx).
): A§
dx√x
=dy√y
=dz
z,
75
1nÙ p©§Ú§|)nØ
kü¼êÕáÄgÈ©
φ1(x, y, z) =√x−√y, φ2(x, y, z) = 2
√y − ln |z|.
©§Ï)
Φ(√x−√y, 2
√y − ln |z|) = 0, (3.3.17)
Ù¥ Φ ´?¿ëY¼ê, ¹k1C.
é¼ê§ (3.3.17) $^Û¼ê3½n)
2√y − ln |z| = g(
√x−√y).
l k
z = e2√yψ(√x−√y).
|^Щ^
ψ(√x− 1) = e−2 cos(ωx).
k
ψ(ζ) = e−2 cos(ω(1 + ζ)2)
ÏdЯK)´
z = e2(√y−1) cos(ω(1 +
√x−√y)2).
2. ¦) ©§
x∂u
∂x+ y
∂u
∂y+ z
∂u
∂z= u+
xy
z.
): ÄA§
dx
x=dy
y=dz
z=
du
u+ xyz
.
´§kü¼êÕáÄgÈ©
φ1(x, y, z, u) =z
x, φ2(x, y, z, u) =
y
z.
duA§È©Ñ u³¡þ, Ïd¦Ù§¼êÕáÄgÈ©±
ò zx
yz ~êw.
76
§3.3 ©§ÈnØ
3A§¥, Ä
dx
x=
du
u+ xyz
,
¿- yz = c1, Kk
du
dx=
1
xu+ c1.
ù´'u u 5©§, §kÏ)
u = x(c1 ln |x|+ c).
Ï A§kÄgÈ©
φ3(x, y, z, u) =u
x− y
zln |x|.
N´y φ1, φ2, φ3 ´¼êÕá. ¤±§Ï)´
Φ( zx,y
z,u
x− y
zln |x|
)= 0,
Ù¥ Φ(ξ, η, ζ) ´?¿ëY¼ê, Φζ Øðu". |^Û¼ê3½n,
§Ï)¤
u = xψ( zx,y
z
)+xy
zln |x|,
Ù¥ ψ ´?¿ëY¼ê.
§3.3.3 Hamilton XÚÈ
È Hamilton XÚäk4Ù´LSN, §Ø=éX²;©Û, éXêAÛ
[52], Riemann AÛÚêÿÀ [7, 8], "AÛÚ"ÿÀ [5, 63] . !0Ù¥f
wÄ:£.
H(q,p) ´ 2n m¥ëY¼ê, Ù¥ q = (q1, . . . , qn), p = (p1, . . . , pn).
©§|
dqi(t)
dt=∂H
∂pi,
dpi(t)
dt= −∂H
∂qi, i = 1, . . . , n,
¡ n gdÝ Hamilton XÚ, H ¡ Hamilton ¼ê. Hamilton ¼êo´A
Hamilton XÚÄgÈ©. þã Hamilton XÚ^Ý/ª¤ q
p
= J∇H, J =
0 E
−E 0
, (3.3.18)
77
1nÙ p©§Ú§|)nØ
Ù¥ 0 ´ n "Ý, E ´ n ü Ý, ∇H L« H FÝ, =
∇H = (Hq1 , . . . ,Hqn , Hp1 , . . . ,Hpn)T ,
Ù¥ T L«Ý=.
±þ½Â´ R2n ¥IO Hamilton XÚ. R2n ¥ Hamilton XÚ´ÏL"
ÝOÝ J ½^ Poisson )Ò5½Â Hamilton XÚ. ù£ÑÖ, Ø3
dQã. k,Ööë [1, 4, 6, 7, 44].
Hamilton XÚþ/Ñy3¢SåÆXÚ¥, ´cÄåXÚÌïÄé.
~K:
1. gdáN$ħ x(t) = g ÏLC q = x, p = mx =z Hamilton XÚ
q =∂H
∂p, p = −∂H
∂q,
Ù¥ Hamilton ¼ê H(q, p) = 12mp
2 −mgq dÄUÚå³U¤, Ù¥ m ´:
þ.
2. ü$ħ x(t) = −a2 sinx (a2 = g/l) ÏLC q = x, p = mx =z Hamilton X
Ú
q =∂H
∂p, p = −∂H
∂q,
Ù¥ Hamilton ¼ê H(q, p) = 12mp
2 − a2m cos q, Ù¥ m ´:þ.
3. üN¯K.^ S L«, E L«/¥,§þ©O´ mS Ú mE . S u.5X
¥%,^ q = (q1, q2, q3) L«.5XI.^ p = (p1, p2, p3) = (mE q1,mE q2,mE q3)
L«Äþ. 3X¥, éuÙ§(¥é/¥^åé, üå±
Ñ. K/¥7$=oUþdÄUÚÚå³U¤
H(q,p) =1
2mE(p2
1 + p22 + p2
3)− GmSmE√q21 + q2
2 + q23
, Ù¥ G ´kÚå~ê.
¤±A Hamilton XÚ
q1 =1
mEp1, p1 = − GmSmEq1
(q21 + q2
2 + q23)
32
,
q2 =1
mEp2, p2 = − GmSmEq2
(q21 + q2
2 + q23)
32
,
q3 =1
mEp3, p3 = − GmSmEq3
(q21 + q2
2 + q23)
32
.
(3.3.19)
78
1nÙ p©§Ú§|)nØ
N´yT Hamilton XÚkÄgÈ©
H1 = q3p2 − q2p3, H2 = q1p3 − q3p1, H3 = q2p1 − q1p2.
du3/¥$1;þzÄgÈ©Ñ~, P
q3(t)p2(t)− q2(t)p3(t) =: c1, q1(t)p3(t)− q3(t)p1(t) =: c2, q2(t)p1(t)− q1(t)p2(t) =: c3.
K
c1q1(t) + c2q2(t) + c3q3(t) ≡ 0.
ùÒy²/¥$1;©ª u.5X¥ÏLI:²¡þ.
e¡?ØBå, Øb/¥3 q3 = 0 ²¡þ$1. l p3 = 0. 3
/¥$1;þ
H(q(t),p(t)) =: c.
du H3(q(t),p(t)) = c3, Kk
q21 + q2
2 − 2µ√q21 + q2
2 = ν, q2q1 − q1q22 = ν1,
Ù¥ ν = 2cm−1E , ν1 = c3m
−1E . |^ÎIC q1 = r cos θ, q2 = r sin θ, ±9 x2 + y2 =
r2 + r2θ2
r2 + r2θ2 − 2µ
r= ν, r2 dθ
dt= −ν1.
k
r = ±√ν + 2µr−1 − ν2
1r−2, θ = −ν1r
−2,
Ù¥ µ = GmS . Øb ν1 > 0, éT§È©
r =ρ
1 + e cos(θ − θ0), ρ =
ν21
µ, e =
ν1
µ
√ν +
µ2
ν21
, (3.3.20)
Ù¥ θ0 ´È©~ê. d²¡)ÛAÛ£, 4I¼ê (3.3.20) 0 < e < 1 ´ý
, e = 1 ´Ô, e > 1 ´V. Ïd/¥$1;´ý.
þã~fL²vê8ÄgÈ©35é Hamilton XÚÄåÆn)å~
Ï. 3²;åÆ¥, Hamilton XÚ (3.3.18) ¡È (½ Liouville È), XJ
• §k n ¼êÕáÄgÈ© H1 = H(q,p), H2(q,p), . . . , Hn(q,p),
79
§3.4 SKn
• ù n ÄgÈ©üüéÜ, = (∇Hi)TJ∇Hj ≡ 0, 1 ≤ i, j ≤ n.
Liouville–Arnold ½nÑÈ Hamilton XÚx. duVyêÆ
õVgÚâ, ÖòØѧQãÚy². k,Ööë [4, 17, 21, 63].
§3.4 SKn
1. y²·K 21.
2. y²½n 22.
3. y²íØ 24.
4. b f(x, y) ∈ C1(Ω), Ω ⊂ R2 ´m«. e φ(x, x0, y0), x ∈ J ´Ð¯K
y′ = f(x, y), y(x0) = y0,
), K
∂φ(x, x0, y0)
∂x0= −f(x0, y0)e
∫ xx0fy(s,φ(s,x0,y0))ds
, x ∈ J,
∂φ(x, x0, y0)
∂y0= e
∫ xx0fy(s,φ(s,x0,y0))ds
, x ∈ J.
5. b y = φ(x, x0, y0) ´Ð¯K
y′ = sin(xyλ), y(x0) = y0,
). Á¦
∂φ(x, x0, y0)
∂λ
∣∣∣∣x0=0,y0=0
,∂φ(x, x0, y0)
∂x0
∣∣∣∣x0=0,y0=0
,∂φ(x, x0, y0)
∂y0
∣∣∣∣x0=0,y0=0
.
6. b p(x), q(x), f(x) 3«m |x − x0| < r þФÂñ?ê. K©§Ð
¯K
y′′ + p(x)y′ + q(x)y = f(x), y(x0) = y0, y′(x0) = y1,
k½Â3 |x− x0| < r þ), T)3 |x− x0| < r þФÂñ?ê.
7. y²·K 28.
80
1nÙ p©§Ú§|)nØ
8. ¦§|
dx
dt= −y + x(x2 + y2)(x2 + y2 − 1),
dy
dt= x+ y(x2 + y2)(x2 + y2 − 1),
ÄgÈ©, ¿(ÜùÄgÈ©½T§|3²¡þ;9Ù5.
9. y²½n 29 N5 1).
10. y²½n 31 N5 4).
11. Φ1(y), . . . ,Φk(y), k < n− 1, ´g£©§|
y′(x) = f(y),
3m« Ω ⊂ Rn þëY¼êÕáÄgÈ©. Φ(y) ´T§|3 Ω þ
,ëYÄgÈ©,
∇yΦ(x,y) = c1(y)∇yΦ1(x,y) + . . .+ cn(y)∇yΦn(x,y).
Ù¥
∇yΦ(x,y) =
(∂Φ(x,y)
∂y1, . . . ,
∂Φ(x,y)
∂yn
),
L« Φ 'u y FÝ. Áyþª¥Xê¼ê ci(y) eØ´~ê, K7´ÄgÈ©.
13. ¦e ©§Ï)
13.1. x∂u∂x + y ∂u∂y + z ∂u∂z = 0;
13.2. (x− z)∂u∂x + (y − z)∂u∂y + 2z ∂u∂z = 0;
13.3. 2x∂u∂x + (y − x)∂u∂y − x2 ∂u∂z = 0;
13.4. x ∂z∂x + 2y ∂z∂y = x2y + z;
13.5. (x2 + y2) ∂z∂x + 2xy ∂z∂y = −z2;
13.6. (y + z)∂u∂x + (z + x)∂u∂y + (x+ y)∂u∂z = u.
14. ¦e ©§÷v½^)
14.1. ∂u∂x + (2ex − y)∂u∂y = 0, u|x=0 = y;
81
§3.4 SKn
14.2.√x∂u∂x +
√y ∂u∂y + z ∂u∂z = 0, u|x=1 = yz + 1;
14.3. x∂u∂x + y ∂u∂y + xy ∂u∂z = z − x2 − y2, u|z=0 = x2 + y2;
14.4. xy ∂z∂x + x2 ∂z∂y = y, z|y=0 = x2;
14.5. xz ∂z∂x + yz ∂z∂y = −xy, y = x2, z = x3;
14.6. x ∂z∂x + y ∂z∂y = z − x2 − y2, x = −2, z = y − y2;
14.7. z ∂z∂x + (z2 − x2) ∂z∂y = −x, y = x2, z = 2x.
15. <E/¥¥(Ï~37/¥,²¡þ$Ä. Á¦<E/¥¥(7/¥$1
;.
82
1oÙ 5©§ÄnØÚ)
Ù8¥?Ø5©§|Úp5©§)nØÚ).
§4.1 5©§)ÄnØ
Ä5©§|
dy
dx= A(x)y + f(x), x ∈ J := (α, β) ⊂ R, (4.1.1)
Ù¥ A(x) = (aij(x))n×n ´ n ,
y =
y1
...
yn
, f(x) =
f1(x)
...
fn(x)
.
• XJ f(x) 6≡ 0, x ∈ J , ¡ (4.1.1) n àg5©§|.
• XJ f(x) ≡ 0, x ∈ J , =
dy
dx= A(x)y, (4.1.2)
¡ n àg5©§|.
éuXþ5©§, ÏL¦)5©§)3Xê¼êëY«m
þÑ3ëY.!ò?Ø5©§|)3«m. Öy²ò^3éõêÆ
Æ¥2¦^ Gronwall ت. §kéõØÓ/ª (k,Öö±ë[??]),
e¡´Ù¥Ä:.
·K34. b c(t), φ(t), g(t) ´ [a, b] þëY¼ê, g(t), c(t) ≥ 0, t ∈ [a, b]. e
φ(t) ≤ c(t) +
∫ t
a
g(s)φ(s)ds, t ∈ [a, b], (4.1.3)
K
φ(t) ≤ c(t)e∫ tag(s)ds, t ∈ [a, b]. (4.1.4)
83
1oÙ 5©§ÄnØÚ)
y: -
Φ(t) = c(t) +
∫ t
a
g(s)φ(s)ds.
K
Φ′(t) = c′(t) + g(t)φ(t) ≤ c′(t) + g(t)Φ(t).
l k (Φ(t) e−
∫ tag(s)ds
)′≤ c′(t)e−
∫ tag(s)ds ≤ c′(t).
ü>l a t È©§
Φ(t) e−∫ tag(s)ds − Φ(a) ≤ c(t)− c(a).
du Φ(a) = c(a), ¤±dþªÚ (4.1.3) = (4.1.4). y..
§4.1.1 5©§|)3«m
½n35. A(x), f(x) ∈ C(J), (x0,y0) ∈ J × Rn. K©§| (4.1.1) ÷vЩ^
y(x0) = y0 )3 J þ3!ëY.
y: du©§| (4.1.1) mà¼ê F(x,y) = A(x)y + f(x) 3 J × Rn þëY,
'u y Jacobi Ý A(x) ëY, ¤±d½n 22 , ЯK)3¹ x0 ,«m
I = (α0, β0) ⊂ J þ3!ëY, Ù¥ I ´3«m. P y = φ(x) ´T). XJ I 6= J ,
Ø β0 < β.
l φ(x) ´§| (4.1.1) )
φ(x)− φ(x0) =
∫ x
x0
A(s)φ(s)ds+
∫ x
x0
f(s)ds, x ∈ [x0, β0), (4.1.5)
du A(x), f(x) 3 [x0, β0] þëY, ¤±
A0 := max1≤i,j≤n
maxx∈[x0,β0]
|aij(x)| <∞, B0 := max1≤i≤n
maxx∈[x0,β0]
|fi(x)| <∞.
- ψ(x) = ‖φ(x)‖ := |φ1(x)|+ . . .+ |φn(x)|. l (4.1.5) , φ(x) z©þ φi(x) Ñ÷v
|φi(x)| ≤ A0
∫ x
x0
ψ(s)ds+B0(β0 − x0) + |φi(x0)|, x ∈ [x0, β0).
ψ(x) ≤ c+
∫ x
x0
nA0 ψ(s)ds, x ∈ [x0, β0),
84
§4.1 5©§)ÄnØ
Ù¥ c = nB0(β0 − x0) + ψ(x0) ≥ 0. l d Gronwall ت
ψ(x) ≤ cenA0(x−x0), x ∈ [x0, β0),
Ïd
limx→β−0
‖φ(x)‖ = limx→β−0
ψ(x) <∞.
ù I = (α0, β0) ´§ (4.1.1) L (x0,y0) ) φ(x) 3«mgñ. ùgñ`²§
(4.1.1) L (x0,y0) )3«m J . y..
§4.1.2 5©§|Ï)(
½Â3«m J þ n þ¼ê y1(x), . . . ,yk(x), k ∈ N,
• ¡3 J þ5', XJ3Ø"~ê c1, . . . , ck ¦
c1y1(x) + . . .+ ckyk(x) ≡ 0, x ∈ J.
• ¡3 J þ5Ã', XJ
c1y1(x) + . . .+ ckyk(x) ≡ 0, x ∈ J,
7k c1 = . . . = ck = 0.
e¡(JÑ5©§|Ï)(.
½n36. b A(x), f(x) 3m«m J þëY, e(ؤá.
(a) àg5©§| (4.1.2))N¤8Ü3þ\Úêþ¦e
¤ n 5m.
(b) y1(x), . . . , yn(x) ´àg5©§| (4.1.2) 5Ã'), y∗(x) ´àg
5©§| (4.1.1) ), K
(b1) àg5©§| (4.1.2) Ï)
y(x) = c1y1(x) + . . .+ cnyn(x), x ∈ J, (4.1.6)
Ù¥ c1, . . . , cn ´?¿~ê, ¹ (4.1.2) ¤k).
85
1oÙ 5©§ÄnØÚ)
(b2) àg5©§| (4.1.1) Ï)
y(x) = c1y1(x) + . . .+ cnyn(x) + y∗(x), x ∈ J, (4.1.7)
Ù¥ c1, . . . , cn ´?¿~ê, ¹ (4.1.1) ¤k).
y: (a). ^ S L«àg5©§| (4.1.2) )N¤8Ü. é ∀y1(x),y2(x) ∈
S, ∀ c1, c2 ∈ R, N´y c1y1(x) + c2y2(x) ∈ S, = S ´5m.
e¡Äky² S ¹k n 5Ã', =§| (4.1.2) 3 J þk n 5Ã'
). ½ x0 ∈ J , ej , j = 1, . . . , n ´1 j ©þ 1 , Ù§Ñ" n ü þ.
d½n 35, àg§| (4.1.2) ÷vЩ^ y(x0) = ej )3 J þ3ëY,
P yj(x), x ∈ J , j = 1, . . . , n. K y1(x), . . . , yn(x) 3 J þ5Ã'. ¯¢þ, XJk
c1y1(x) + . . .+ cnyn(x) ≡ 0, x ∈ J,
K
c1e1 + . . .+ cnen = c1y1(x0) + . . .+ cnyn(x0) = 0.
w,ùàg5ê§|k) c1 = . . . = cn = 0. ùÒy² S ¥k n 5
Ã')þ.
Ùgy² S ¥?±d y1(x), . . . , yn(x) 3 J þ5L«. y(x) ´ (4.1.2)
3 J þ?). - y0 = y(x0). c1, . . . , cn ´àg5ê§|
c1e1 + . . .+ cnen = y0,
). Kd)5
y(x) = c1y1(x) + . . .+ cnyn(x), x ∈ J,
ù´Ï§Ñ´àg5©§| (4.1.2) )÷vÓЩ^. ùÒy² S
´ n 5m.
(b). (b1) y²d (a) N´.
(b2) y, é?¿~ê c1, . . . , cn, ¼ê
Φ(x, c) = c1y1(x) + . . .+ cnyn(x) + y∗(x), x ∈ J,
86
§4.1 5©§)ÄnØ
´àg5©§| (4.1.1) ), Ù¥ c = (x1, . . . , cn). q∂Φ
∂c= (y1(x), . . . , yn(x)),
¤±~êþ c ´Õá. Ï Φ(x, c) ´5©§| (4.1.1) Ï).
ey
c1y1(x) + . . .+ cnyn(x) + y∗(x); x ∈ J, c1, . . . , cn ´?¿~ê,
¹5©§| (4.1.1) ¤k). y(x), x ∈ J ´§| (4.1.1) ). K
y(x)− y∗(x) ´àg5©§| (4.1.2) ). d (a) 3 c1, . . . , cn ¦
y(x)− y∗(x) = c1y1(x) + . . .+ cnyn(x), x ∈ J.
ùÒy² (4.1.7) ¹àg5©§| (4.1.1) ¤k). y..
½n 36 ( n àg©§| (4.1.2) n 5Ã')35. àg5
©§| (4.1.2) ?¿ n 5Ã')¡§Ä)|.
XÛ^'½ n )´Ä5Ã'?
y1(x) =
y11
...
yn1
, . . . , yn(x) =
y1n
...
ynn
, (4.1.8)
´ (4.1.2) n ).
• ݼê Y(x) = (yij)1≤i,j≤n ¡§| (4.1.2) )Ý.
• XJ y1(x), . . . ,yn(x) 5Ã', ¡ Y(x) §| (4.1.2) Ä)Ý.
• )Ý1ª det Y(x) ¡)| (4.1.8) Wronsky 1ª, P W (x).
·K37. 'uàg5©§| (4.1.2) )| (4.1.8), e(ؤá.
(a) ©§| (4.1.2) )| (4.1.8) 5Ã'= W (x) 6= 0, x ∈ J ;
(b) ©§| (4.1.2) )| (4.1.8) 5'= W (x) ≡ 0, x ∈ J .
y: Äkyé ∀x0 ∈ J k
W (x) = W (x0)e∫ xx0
trA(s)ds, ¡ Liouville úª,
87
1oÙ 5©§ÄnØÚ)
Ù¥ trA(x) = a11(x) + . . .+ ann(x) ݼê A(x) ,. ¯¢þ, é W (x) ¦ê, ¿|
^1ª¦K
dW (x)
dx=∑
1≤i≤n
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
y11 · · · y1n
... · · ·...
n∑j=1
aijyj1 · · ·n∑j=1
aijyjn
.... . .
...
yn1 · · · ynn
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= (trA(x))W (x).
éþã'u W Cþ©l§l x0 x È©, = Liouville úª.
·K (a) Ú (b) (Øl Liouville úªN´. y..
~K: ^·K 37 ½e¼ê| 1
x
,
x
x3
, (4.1.9)
´Ä´,àg5©§|3 R þü).
): Ï
W (x) =
∣∣∣∣∣∣∣1 x
x x3
∣∣∣∣∣∣∣ = x3 − x2,
3 RQk":k":, Ïd (4.1.9)¥¼ê|ØU´?Ûàg5©§|
3 R þ).
e¡(ØÑàg5©§| (4.1.2) Ä)Ý5.
íØ38. Φ(x) ´§| (4.1.2) Ä)Ý, e(ؤá:
(a) y(x) = Φ(x)c ´ (4.1.2) Ï), Ù¥ c ∈ Rn ´?¿~êþ;
(b) C ´ n ÛÉ~êÝ, K Φ(x)C ´ (4.1.2) Ä)Ý;
(c) e Ψ(x) ´ (4.1.2) Ä)Ý, K3ÛÉ n ~êÝ C ¦ Ψ(x) =
Φ(x)C.
½n 36Ñàg©§| (4.1.1)Ï)dàg©§| (4.1.2)Ï)à
g©§| (4.1.1) ?)Ú¤. e¡½nÑXÛlàg5©§|
Ï)àg5©§|Ï).
88
§4.1 5©§)ÄnØ
½n39. Φ(x) ´àg5©§| (4.1.2) Ä)Ý, Kàg5©§|
(4.1.1) Ï)
y(x) = Φ(x)
(c +
∫ x
x0
Φ−1(s)f(s)ds
), (4.1.10)
Ù¥ c ∈ Rn ´?¿~êþ.
y: ±y (4.1.10)´ (4.1.1)), c´Õá~êþ, Ï (4.1.10) ´àg
5©§| (4.1.1) Ï).
e¡Ñ (4.1.10) íL§. - y∗(x) = Φ(x)c(x), òÙ\§| (4.1.1), ¿z
Φ(x)c′(x) = f(x).
éT§l x0 x È©
c(x) =
∫ x
x0
Φ−1(s)f(s)ds+ c,
Ù¥ c ´È©~ê. §| (4.1.1) k)
y∗(x) = Φ(x)
∫ x
x0
Φ−1(s)f(s)ds,
ÚÏ) (4.1.10). y..
N5:
1. 3½n 39 y²L§¥, ÏLòàg§| (4.1.2) Ï)¥?¿~êC¤'u
x ¼ê, l ¦Ñàg§| (4.1.1) Ï). ù«¡~êC´. Ï)úª
(4.1.10) ¡~êC´úª.
2. dÏ)úª (4.1.10)N´¦§| (4.1.1)÷vЩ^ y(x0) = y0 ЯK)
~êC´úª
y(x) = Φ(x)
(Φ−1(x0)y0 +
∫ x
x0
Φ−1(s)f(s)ds
).
3. ~êC´úª$^u5©§|. A(x) 3 J þëY, Φ(x) ´ (4.1.2)
Ä)Ý. XJ f(x,y) 3 J × Rn þëY, KЯK
y′ = A(x)y + f(x,y), y(x0) = y0,
89
1oÙ 5©§ÄnØÚ)
) y = φ(x) ÷vÈ©§
φ(x) = Φ(x)
(Φ−1(x0)y0 +
∫ x
x0
Φ−1(s)f(s, φ(s))ds
).
þã½nL², ¦)àg5©§|, 7LÄkÙAàg5
©§|Ä)Ý. 5`, ¦)àg5©§|Ä)Ý´~(J.
e!òÑ~Xêàg5©§|Ä)ݦ. e¡Þ¦)àg5©
§|ü~f.
~K: ¦)eàg5©§|ЯK:
x′(t) = −2
tx+ 1, x(1) =
1
3,
y′(t) =
(1 +
2
t
)x+ y − 1, y(1) = −1
3.
): làg§
x′(t) = −2
tx, y′(t) =
(1 +
2
t
)x+ y.
1§k)Ñ x = c1t−2, 2\1§)Ñ y = −c1t−2 + c2e
t. l àg
§|Ä)Ý
Φ(t) =
t−2 0
−t−2 et
§) x
y
= Φ(t)
Φ−1(1)
3−1
−3−1
+
∫ t
1
s2 0
e−s e−s
1
−1
ds
=
13 t
− 13 t
.
§4.1.3 p5©§Ï)(
¦+p5©§±=z¤5©§|, dup5©§g
A:, ±$^B, !üÕ?Ø.
n 5©§
y(n) + a1(x)y(n−1) + . . .+ an−1(x)y′ + an(x)y = f(x), x ∈ J, (4.1.11)
• ¡àg, XJ f(x) 6≡ 0.
90
§4.1 5©§)ÄnØ
• ¡ àg, XJ f(x) ≡ 0, =
y(n) + a1(x)y(n−1) + . . .+ an−1(x)y′ + an(x)y = 0, x ∈ J. (4.1.12)
p§ (4.1.11) 3C y1 = y(x), y2 = y′(x), . . . , yn = y(n−1)(x) ez§|
dy
dx= A(x)y + f(x), (4.1.13)
Ù¥
A(x) =
0 1 0 · · · 0
0 0 1 · · · 0
......
.... . .
...
0 0 0 · · · 1
−an(x) −an−1(x) −an−2(x) · · · −a1(x)
,
y(x) =
y1
y2
...
yn−1
yn
, f(x) =
0
0
...
0
f(x)
.
l n 5§ (4.1.11) n 5§| (4.1.13) m'XN´,
• y = φ(x) ´ (4.1.11) )= y = (φ(x), φ′(x), . . . , φ(n−1)(x))T ´ (4.1.13) ),
Ù¥ T L«þ=.
aq/, éuàg5§ (4.1.12) ) φ1(x), . . . , φn(x) ±½Â§ Wronsky 1
ª
W (x) =
∣∣∣∣∣∣∣∣∣∣∣∣∣
φ1(x) φ2(x) · · · φn(x)
φ′1(x) φ′2(x) · · · φ′n(x)
......
. . ....
φ(n−1)1 (x) φ
(n−1)2 (x) · · · φ
(n−1)n (x)
∣∣∣∣∣∣∣∣∣∣∣∣∣,
÷v
W (x) = W (x0)e−
∫ xx0a1(s)ds
, x ∈ J, ¡ Liouville úª.
2l5©§|)nØ·±e¡(Ø.
91
1oÙ 5©§ÄnØÚ)
½n40. éu n àg5©§ (4.1.11) Úàg5©§ (4.1.12), e(ؤ
áµ
(a) b a1(x), . . . , an(x), f(x) ∈ C(J), Ké ∀ (x0, y0, y1, . . . , yn−1) ∈ J × Rn, § (4.1.11)
÷vЩ^
y(x0) = y0, y′(x0) = y1, . . . , y
(n−1)(x0) = yn−1,
)3 J þ3ëY.
(b) φ1(x), . . . , φn(x) ´àg§ (4.1.12) ). K§3 J þ5Ã'¿^´
Wronsky 1ª W (x) 6= 0, x ∈ J . d¡ φ1(x), . . . , φn(x) àg§ (4.1.12) Ä
)|.
(c) φ1(x), . . . , φn(x) ´àg§ (4.1.12) Ä)|, K
(c1) àg§ (4.1.12) Ï)
y(x) = c1φ1(x) + . . .+ cnφn(x), x ∈ J,
Ù¥ c1, . . . , cn ´?¿~ê;
(c2) àg§ (4.1.11) Ï)
y(x) = c1φ1(x) + . . .+ cnφn(x) + φ∗(x), x ∈ J,
Ù¥ c1, . . . , cn´?¿~ê, φ∗´ (4.1.11)?).AO/, φ∗d φ1(x), . . . , φn(x)
L«, =
φ∗(x) =
n∑k=1
φk(x)
∫ x
x0
Wk(s)
W (s)f(s)ds,
Wk(x) ´ W (x) (n, k) êfª, =
Wk(x) = (−1)n+k
∣∣∣∣∣∣∣∣∣∣φ1(x) · · · φk−1(x) φk+1(x) · · · φn(x)
......
......
......
φ(n−2)1 (x) · · · φ
(n−2)k−1 (x) φ
(n−2)k+1 (x) · · · φ
(n−2)n (x)
∣∣∣∣∣∣∣∣∣∣.
92
§4.1 5©§)ÄnØ
y: (a), (b), (c1) y²±d§|)nØN´, ÖögC¤. ey (c2). -
Φ(x) =
φ1(x) φ2(x) · · · φn(x)
φ′1(x) φ′2(x) · · · φ′n(x)
......
. . ....
φ(n−1)1 (x) φ
(n−1)2 (x) · · · φ
(n−1)n (x)
.
K§| (4.1.13) Ï)
y(x) =
y(x)
y′(x)
...
y(n−1)(x)
= Φ(x)c + Φ(x)
∫ x
x0
Φ−1(s)f(s)ds.
e¡¦ y(x) Lª. w, Φ(x)c 11 c1φ1(x) + . . .+ cnφn(x). d_Ý
êfªL«
Φ−1(s) =1
W (s)
∗ · · · ∗ W1(s)
.... . .
......
∗ · · · ∗ Wn(s)
,
Ù¥ ∗ ´vkÑwªLªþ©l k
Φ−1(s)f(s) =1
W (s)
f(s)W1(s)
...
f(s)Wn(s)
,
Φ(x)∫ xx0
Φ−1(s)f(s)ds 11 φ∗(x). y.©
N5:
1. n = 2 , § (4.1.11) Ï)±¤
y(x) = c1φ1(x) + c2φ2(x) +
∫ x
x0
φ1(s)φ2(x)− φ2(s)φ1(x)
φ1(s)φ′2(s)− φ2(s)φ′1(s)f(s)ds. (4.1.14)
2. n àg§ (4.1.11) Ï)±làg§ (4.1.12) Ï)ÏL~êC´
µ-
φ(x) = c1(x)φ1(x) + . . .+ cn(x)φn(x).
93
1oÙ 5©§ÄnØÚ)
é φ(x) Åg¦ê, 3 φ(k)(x), k = 1, . . . , n− 1, ¥-
c′1(x)φ(k−1)1 (x) + . . .+ c′n(x)φ(k−1)
n (x) = 0, k = 1, . . . , n− 1, (4.1.15)
¿ò φ(k)(x), k = 0, 1, . . . , n, \§ (4.1.11) ÏLz
c′1(x)φ(n−1)1 (x) + . . .+ c′n(x)φ(n−1)
n (x) = f(x). (4.1.16)
l n àg5ê§ (4.1.15) Ú (4.1.16) ¥)Ñ c′1(x), . . . , c′n(x), ¿È©=
(4.1.11) Ï).
~K: y = φ(x) ´§
y′′ + p(x)y′ + q(x)y = 0, x ∈ (a, b) (4.1.17)
"), p(x), q(x) 3 (a, b) þëY, K
(a) φ(x)3 (a, b)?¿4f«mþõkk":, 3 φ(x)":?ÙêØ";
(b) § (4.1.17) Ï)
y(x) = φ(x)
(c1 + c2
∫ x
x0
1
φ2(s)e−
∫ sx0p(t)dt
ds
),
Ù¥ c1, c2 ´?¿~ê. XJ3 x0, x mk φ(s) ":, Ø x0 < x1 < . . . <
xk < x, KþãÈ©´d©ãÈ©½Â.
y: db, (4.1.17) )3 (a, b) þëY.
(a) y. XJ φ(x) 3 (a, b) ,4f«mþkáõ":, K φ(x) ù":7
kà:, P x∗. xn ´ φ(x) ":, limn→∞
xn = x∗. K
φ(x∗) = limn→∞
φ(xn) = 0, φ′(x∗) = limn→∞
φ(xn)− φ(x∗)
xn − x∗= 0.
d)5 φ(x) ≡ 0, bgñ. l φ(x) 3 (a, b) ?Û4f«mþõkk
":.
XJ x∗ ´ φ(x) ":, K φ′(x∗) 6= 0. ÄKd)5 φ(x) ≡ 0, x ∈ (a, b), gñ.
(b) Äkb φ(x) 6= 0, x ∈ (a, b). y(x) ´§ (4.1.17) ?). Kd Liouville úª
φ(x)y′(x)− y(x)φ′(x) = ce−
∫ xx0p(t)dt
=⇒ d
dx
(y
φ
)=
c
φ2e−
∫ xx0p(t)dt
.
94
§4.2 ~Xê5©§|)
y
φ= c
∫ x
x0
1
φ2(s)e−
∫ sx0p(t)dt
ds+ c1.
ùÒ§ (4.1.17) Ï).
XJ φ(x) k":, x∗ ´l x0 C":. Ø x0 < x∗ x0 Ø´":. K
4
limx→x−∗
φ(x)
∫ x
x0
1
φ2(s)e−
∫ sx0p(t)dt
ds = limx→x−∗
φ−2(x)e−
∫ xx0p(t)dt
−φ−2(x)φ′(x)= −e
−∫ x∗x0
p(t)dt
φ′(x∗),
3. Ï é ∀x ∈ (a, b), XJ x1, . . . , xk ´ φ(x) 3 x0 x m":, Ø x0 <
x1 < . . . < xk < x, K¼ê
y(x) = φ(x)
(c1 + c2
(∫ x1
x0
+ . . .+
∫ xk
xk−1
+
∫ x
xk
)1
φ2(s)e−
∫ sx0p(t)dt
ds
),
´k½Â, § (4.1.17) Ï). y..
N5:
• aquCXêàg5©§|, pCXêàg5©§vk).
XJÙ"), Kp§±ü$. öSÖögC¤.
• Euler §
xndny
dxn+ a1x
n−1 dn−1y
dxn−1+ . . .+ an−1x
dy
dx+ any = 0,
Ù¥ a1, . . . , an ∈ R ´~ê, ±ÏLgCþC x = et z~Xê5©§.
eü!?Ø~Xê5©§).
§4.2 ~Xê5©§|)
Ä~Xê5©§|
dy
dx= Ay + f(x), x ∈ J = (a, b), (4.2.1)
Ù¥ A ´ n ¢~êÝ, f(x) ∈ C(J). n = 1 , P A = a, § (4.2.1) éAàg
§kÏ) y = ceax. Á n > 1 àg5©§| (4.2.1) éAàg5©
§|Ï)´Äkaq/ª?
95
1oÙ 5©§ÄnØÚ)
§4.2.1 Ýê¼ê~Xê5©§|)
^ML« n¢~êÝN¤8Ü.KM3Ý\ÚÝ¢ê¦
e¤5m. é A = (aij) ∈M, ½Â A
‖A‖ =
n∑i,j=1
|aij |.
KÝ÷ve5µé ∀A,B ∈M, ∀λ ∈ R,
1) ‖A‖ ≥ 0, ‖A‖ = 0 ⇐⇒ A = 0;
2) ‖λA‖ = |λ|‖A‖, λ ∈ R;
3) ‖A + B‖ ≤ ‖A‖+ ‖B‖;
4) ‖AB‖ ≤ ‖A‖ ‖B‖, ‖Ak‖ ≤ ‖A‖k, k ∈ N.
c 3 ^5´w,. ey1 4 ^µé A = (aij), B = (bij), - cij =n∑k=1
aikbkj . K
AB = (cij). l
‖AB‖ =
n∑i,j=1
|cij | ≤n∑i=1
n∑j=1
n∑k=1
|aik||bkj | =n∑i=1
n∑k=1
|aik|n∑j=1
|bkj |
≤n∑i=1
n∑k=1
|aik|n∑
k,j=1
|bkj | = ‖A‖ ‖B‖.
?Ú/, ‖Ak‖ ≤ ‖Ak−1‖ ‖A‖ ≤ ‖A‖k.
·K41. é ∀A,B ∈M, e(ؤáµ
(a) Ý?ê
E + A +1
2!A2 + . . .+
1
k!Ak + . . . ,
ýéÂñ. PÙ eA ½ exp(A), ¡Ýê¼ê;
(b) XJ A,B , = AB = BA, K eA+B = eAeB;
(c) é?¿ A ∈M, K eA _, (eA)−1
= e−A;
(d) é?¿_Ý P ∈M k ePAP−1
= PeAP−1.
96
§4.2 ~Xê5©§|)
y: (a) P a = ‖A‖. K ∥∥∥∥∥∞∑k=0
Ak
k!
∥∥∥∥∥ ≤∞∑k=0
‖Ak‖k!≤∞∑k=0
ak
k!<∞.
¤±Ý?êýéÂñ, l ∞∑k=0
Ak
k! ∈ M. þãتy²Ý?êz©þ
ÑýéÂñ.
(b) O
eA+B =
∞∑k=0
(A + B)k
k!=
∞∑k=0
k∑i=0
k
i
AiBk−i
k!=
∞∑k=0
k∑i=0
AiBk−i
i!(k − i)!= eAeB,
Ù¥
k
i
=k!
i!(k − i)!.
(c) Ú (d) y²N´, lÑ. y..
½n42. éu~Xê5©§|, e(ؤáµ
(a) Ýê¼ê Y(x) = exA ´~Xêàg5©§|
dY
dx= AY, (4.2.2)
Ä)Ý;
(b) f(x) ∈ C(J), x0 ∈ J , K~Xêàg5©§| (4.2.1)
– Ï)
y(x) = exAc +
∫ x
x0
e(x−s)Af(s)ds,
Ù¥ c ´?¿ n ~êþ.
– L (x0,y0) ∈ J × Rn ⊂ Rn+1 ЯK)
y(x) = e(x−x0)Ay0 +
∫ x
x0
e(x−s)Af(s)ds.
y: (a) é?¿ x ∈ R, - I ⊂ R ´± x S:k.m«m. Ï
exA =
∞∑k=0
xkAk
k!,
97
1oÙ 5©§ÄnØÚ)
3 I þýéÂñ, ¤±éÝê¼êŦ
dexA
dx=
∞∑k=1
xk−1Ak
(k − 1)!= A
∞∑k=1
xk−1Ak−1
(k − 1)!= A exA,
= exA ´àg5©§| (4.2.2) )Ý. q e0A = E, ¤± exA ´àg5©
§| (4.2.2) Ä)Ý.
(b) d (a) Úàg5©§|~êC´úªá. y..
N5:
• ½n 42 lnØþ)û~Xê5©§|¦)¯K.
• éu½ A ∈M, XÛ¦ exA k)û?
§4.2.2 ~Xêàg5©§|Ä)ݦ
1. ^ Jordan IO.¦Ä)Ý
é ∀A ∈M, d5ê Jordan IO.nØ, 3ÛÉÝ P ∈M ¦
A = PJP−1,
Ù¥
J = diag(J1, . . . ,Jm) =
J1
J2
. . .
Jm
,
Ji = λiEni + Ni, Ni =
0 0 0 · · · 0 0
1 0 0 · · · 0 0
0 1 0 · · · 0 0
.... . .
. . .. . .
......
0 0 0. . . 0 0
0 0 0 . . . 1 0
,
Ù¥ λi, i = 1, . . . ,m, ´ A A, ni ´Ý Ji ê, n1 + . . .+ nm = n, Eni ´ ni
ü Ý.
exA = PexJP−1 = P diag(exJ1 , . . . , exJm
)P−1.
98
§4.2 ~Xê5©§|)
qé i = 1, . . . ,m,
exJi = exλiEni exNi = eλixexNi = eλix
1 0 0 · · · 0 0
x 1 0 · · · 0 0
x2
2! x 1 · · · 0 0
......
. . .. . .
......
xni−2
(ni−2)!xni−3
(ni−3)!xni−4
(ni−4)!
. . . 1 0
xni−1
(ni−1)!xni−2
(ni−2)!xni−3
(ni−3)! . . . x 1
.
ùÒlnØþ)ûÄ)Ý exA ¦¯K.
N5: lÝê¼ê¦Ä)Ý, ±9 exAP E´Ä)Ý, àg5©§|
(4.2.2) kÄ)Ý PexJ. w,, §zþ´d eλix gêØL ni − 1 õª
¦È¤. Ä)Ýù«(e¡Ïé#¦)Jøg´.
~K: ¦)e~Xêàg5©§|
dy
dx= Ay + f(x), y =
y1
y2
y3
, A =
2 0 0
0 −1 0
0 1 −1
, f(x) =
0
1
x
,
): O
exA =
e2x 0T
0 e−x exp
x 0 0
1 0
=
e2x 0 0
0 e−x 0
0 xe−x e−x
.
¤±d~êC´úª§Ï)
y(x) = exAc +
∫ x
0
e(x−s)Af(s)ds = exAc +
0
1− e−x
x− xe−x
,
Ù¥ c ´?¿ 3 ~êþ.
¦+Ýê¼ê)û~Xê5©§Ä)ݦ¯K, ¢SO%
´~(J, Ï¦Ý Jordan IO.5Ò´©(J¯. e¡Jø,
´uO.
99
1oÙ 5©§ÄnØÚ)
2. AAþ¦Ä)Ý
ãTI5êe(Ø (ë [35, § 6.1,½n 3])µ
·K43. λ1, . . . , λs ´Ý A¤kpØÓA,§ê©O n1, . . . , ns
n1 + . . .+ ns = n. K
• 5ê§|
(A− λiE)nir(i)0 = 0, (4.2.3)
k ni 5Ã'), P r(i)j0 , j = 1, . . . , ni;
• n þ r(1)10 , . . . , r
(1)n10, . . . , r
(s)10 , . . . , r
(s)ns05Ã'.
½n44. λ1, . . . , λs ´Ý A ¤kpØÓA, §ê©O n1, . . . , ns
n1 + . . . + ns = n. P r(i)10 , . . . , r
(i)ni0àg5ê§| (4.2.3) ni 5Ã'),
i = 1, . . . , s. Ke(ؤá.
(a) XJ s = n, K ni = 1, i = 1, . . . , n, r(i)10 ´ λi Aþ. ݼê
Φ(x) =(eλ1xr
(1)10 , . . . , e
λnxr(n)10
),
´~Xêàg5©§| (4.2.2) Ä)Ý;
(b) XJ s < n, é ni > 1, j = 1, . . . , ni, -
r(i)jl = (A− λiE)r
(i)j,l−1, l = 1, . . . , ni − 1.
Kݼê
Φ(x) =(eλ1xP
(1)1 (x), . . . , eλ1xP(1)
n1(x), . . . , eλsxP
(s)1 (x), . . . , eλsxP(s)
ns (x)),
´~Xêàg5©§| (4.2.2) Ä)Ý, Ù¥
P(i)j (x) =
ni−1∑k=0
xk
k!r
(i)jk , i = 1, . . . , s, j = 1, . . . , ni.
y: (a). duéAØÓAAþ5Ã', ¤±
Φ(0) =(r
(1)10 , . . . , r
(n)10
),
100
§4.2 ~Xê5©§|)
´ÛÉÝ. q
dΦ(x)
dx=(eλ1xλ1r
(1)10 , . . . , e
λnxλnr(n)10
)=(eλ1xAr
(1)10 , . . . , e
λnxAr(n)10
)= AΦ(x).
ùÒy² Φ(x) ´~Xêàg5©§| (4.2.2) Ä)Ý.
(b). d·K 43
Φ(0) =(r
(1)10 , . . . , r
(1)n10, . . . , r
(s)10 , . . . , r
(s)ns0
),
ÛÉ.e¡Iy² eλixP(i)j (x), i = 1, . . . , s, j = 1, . . . , ni ´àg5©§| (4.2.2)
). ¯¢þ,
d
dx
(eλixP
(i)j (x)
)= λie
λixni−1∑k=0
xk
k!r
(i)jk + eλix
ni−1∑k=1
xk−1
(k − 1)!r
(i)jk
= λieλix
ni−1∑k=0
xk
k!r
(i)jk + eλix
ni−1∑k=1
xk−1
(k − 1)!(A− λiE) r
(i)j,k−1
= λieλix
xni−1
(ni − 1)!r
(i)j,ni−1 + eλix
ni−2∑k=0
xk
k!Ar
(i)jk
= Aeλixni−1∑k=0
xk
k!r
(i)jk = AeλixP
(i)j (x),
Ù¥31 4 ª¥^ λir(i)j,ni−1 = Ar
(i)j,ni−1, ù´du
(A− λiE)r(i)j,ni−1 = (A− λiE)2r
(i)j,ni−2 = . . . = (A− λiE)nir
(i)j0 = 0.
ùÒy² Φ(x) ´~Xêàg5©§| (4.2.2) Ä)Ý. y..
N5:
1. ½n 44 (a)¦ npØÓA,Ù¢ly²±µXJ Ak n5Ã
'Aþ r1, . . . , rn, éAA´ λ1, . . . , λn (§kU), K
Φ(x) =(eλ1xr1, . . . , e
λnxrn),
´©§| (4.2.2) Ä)Ý.
2. e¢Ý A kEA λi, K λj = λi ´A (îL«Ý). P r(i)m0, r
(j)m0 = r
(i)m0,
m ∈ 1, . . . , ni©O´5ê§| (4.2.3)éAu λi Ú λj ). KÄ)Ý Φ(x)
´E. ¡±l exA = Φ(x)Φ−1(0) ¢Ä)Ý. ,¡éuàg5
101
1oÙ 5©§ÄnØÚ)
©§| (4.2.2) ?éÝE) eλixP(i)m (x) Ú eλjxP
(j)m (x) = eλixP
(i)
m (x), ±ÏL
-
eλixP(i)m (x) = u(x) +
√−1v(x), eλixP
(i)
m (x) = u(x)−√−1v(x),
àg5©§| (4.2.2) ü¢) u(x) Ú v(x). 2^ùü¢)OÄ)
Ý¥ùéÝE)=¢Ä)Ý.
~K:
1. ¦~Xêàg5©§|
dy
dx= Ay, A =
2 −1 −1
−2 1 3
0 −1 1
,
Ä)Ý.
): Ï det(A−λE) = (2−λ)((λ− 1)2 + 1
),¤±AA λ1 = 2, λ2 = 1+
√−1,
λ3 = 1−√−1. §éAAþ©O
r1 =
2
−1
1
, r2 =
1
−√−1
1
, r3 =
1
√−1
1
.
e(1+√−1)xr2 ½ e(1−
√−1)xr3 ¢ÜÚJܧü5Ã'¢)
ex
cosx
sinx
cosx
, ex
sinx
− cosx
sinx
.
§kÄ)Ý
Φ(x) =
2e2x ex cosx ex sinx
−e2x ex sinx −ex cosx
e2x ex cosx ex sinx
.
102
§4.2 ~Xê5©§|)
2. ¦~Xêàg5©§|
dy
dx= Ay, A =
0 1 0 0
−1 2 0 0
−2 2 1 0
0 1 0 −1
,
Ä)Ý.
): Ï det(A − λE) = (λ + 1)(λ − 1)3, ¤± A A λ1 = −1, λ2 = 1 (n).
λ1 = −1 éAAþ r1 = (0, 0, 0, 1)T .
éu λ2 = 1, l5ê§|
(A−E)3r = 0,
)
r(2)10 =
−1
1
0
0
, r
(2)20 =
0
0
1
0
, r
(2)30 =
4
0
0
1
.
?Ú/,
r(2)11 = (A−E)r
(2)10 =
2
2
4
1
, r
(2)12 = (A−E)r
(2)11 =
0
0
0
0
,
r(2)21 = (A−E)r
(2)20 =
0
0
0
0
,
r(2)31 = (A−E)r
(2)30 =
−4
−4
−8
−2
, r
(2)32 = (A−E)r
(2)31 =
0
0
0
0
.
103
1oÙ 5©§ÄnØÚ)
¤±§kÄ)Ý
Φ(x) =(e−xr1 ex(r
(2)10 + xr
(2)11 ) exr
(2)20 ex(r
(2)30 + xr
(2)31 ))
=
0 −ex + 2xex 0 4ex − 4xex
0 ex + 2xex 0 −4xex
0 4xex ex −8xex
e−x xex 0 ex − 2xex
.
e¡|^½n 44 Ñ~Xêàg5©§|)O.
íØ45. A ∈M. XJ A A¢ÜÑK, Ké ∀v ∈ Rn, ∃ ρ > 0, a > 0 ¦
‖exAv‖2 ≤ ae−ρx‖v‖2, x ≥ 0.
Ù¥ ‖v‖2 =√v2
1 + . . .+ v2n.
y: d½n 44,
Φ(x) =(eλ1xP
(1)1 (x), . . . , eλ1xP(1)
n1(x), . . . , eλsxP
(s)1 (x), . . . , eλsxP(s)
ns (x)),
´§| (4.2.2) Ä)Ý, ¤±3ÛÉÝ C ∈M ¦
exA = Φ(x)C =: (w1(x), . . . ,wn(x)) ,
Ù¥
wi(x) =s∑j=1
Pij(x)eλj x, i = 1, . . . , n,
Pij(x) ´gê ≤ n− 1 n þõª.
P λj = αj +√−1βj , α = max
1≤j≤sαj , KdnØªÚ Cauchy ت
n∑i=1
‖wi(x)‖22 ≤n∑i=1
s∑j=1
‖Pij(x)eλjx‖2
2
=
n∑i=1
s∑j=1
‖Pij(x)‖2eαjx2
≤n∑i=1
s∑j=1
‖Pij(x)‖22s∑j=1
e2αjx ≤ ne2αxn∑i=1
s∑j=1
‖Pij(x)‖22
≤ ne2αxn∑i=1
s∑j=1
nM2
(n−1∑k=0
|x|k)2
≤ n4M2e2αx
(n−1∑k=0
|x|k)2
,
104
§4.2 ~Xê5©§|)
Ù¥ M ´ Pij(x), i = 1, . . . , n, j = 1, . . . , s Xêýé. ¤±2dnت
Ú Cauchy ت
‖exAv‖2 ≤n∑i=1
‖wi(x)‖2 |vi| ≤ ‖v‖2
(n∑i=1
‖wi(x)‖22
) 12
≤ ‖v‖2 n2Mexαn−1∑k=0
|x|k.
ρ ∈ (0,−α). du
limx→∞
ex(α+ρ)n−1∑k=0
|x|k = 0,
¤±3 K0 > 0 ¦
exαn−1∑k=0
|x|k ≤ K0e−ρx, x ∈ [0,∞).
- a = n2MK0 =íØy². y..
§4.2.3 A^µ²¡~Xê5©XÚÛÜ(
~Xêàg5©§|)A^, !?ز¡~Xê5©§|
d
dt
x
y
= A
x
y
, A =
a b
c d
6= 0, (4.2.4)
3Û: (0, 0) ;ÛÜ(, Ù¥ 0 L« 2 "Ý.
éu©§|
dx
dt= P (x, y),
dy
dt= Q(x, y),
• : (x0, y0) ∈ R2 ¡T§|Û:, XJ P (x0, y0) = 0, Q(x0, y0) = 0.
• Û: (x0, y0) ¡ÐÛ: (½pÛ:), XJ P,Q 3 (x0, y0) Jacobi Ý
∂(P,Q)
∂(x, y)
∣∣∣∣(x0,y0)
,
AkØ" (½Ñ").
• XJÐÛ:üAÑØ", ¡òz. ÄK¡òz.
• XJòzÐÛ:A¢ÜÑØ", ¡V. ÄK¡V.
105
1oÙ 5©§ÄnØÚ)
ã 4.1 ½.(:ؽ.(:
d JordanIO.nØ,3¢_Ý P¦ P−1APe JordanIO.µ λ 0
0 µ
,
λ 0
1 λ
,
α −β
β α
,
λ 0
0 0
,
0 0
1 0
,
Ù¥ λ, µ, β 6= 0. N´y3C x
y
= P
ξ
η
,
e, §| (4.2.4) z
d
dt
ξ
η
= P−1AP
ξ
η
.
du5Cå.Ú^=^, ÏdØ5, e¡?Ø A äkþãIO
.§| (4.2.4) 3Û: (0, 0) 5.
(I) A =
λ 0
0 µ
.
àg5©§| (4.2.4) Ï)
x = c1eλt, y = c2e
µt,
Ù¥ c1, c2 ´?¿~ê. ?Ú/, §| (4.2.4) )L«¤
x = 0, ½ y(x) = c|x|µλ ,
Ù¥ c ´?¿~ê.
1) λ = µ. §| (4.2.4) ) x = 0, ½ y = cx, Ù¥ c ´?¿~ê. ©§|
(4.2.4) )3 (0, 0) ÛÜ(Xã 4.1. d (0, 0) ¡.(:. λ = µ < 0
, ¡½.(:; λ = µ > 0 , ¡Ø½.(:.
106
§4.2 ~Xê5©§|)
ã 4.2 ½ü(:ؽü(:
ã 4.3 Q:
2) λ 6= µ, λµ > 0. µ
λ> 1 , ¤k; (Ø x = 0 ) 3:? x ¶.
µ
λ< 1 , ¤k; (Ø y = 0 ) 3:? y ¶. ©§| (4.2.4) )
3 (0, 0) ÛÜ(Xã 4.2. d (0, 0) ¡ü(:, ½¡(:. λ > 0
(½ λ < 0) , ¡Ø½(: (½½(:).
3) λµ < 0. du
limx→±∞
c|x|µλ = 0, lim
x→±0c|x|
µλ =∞,
¤±©§| (4.2.4) )3 (0, 0) ÛÜ(Xã 4.3. d (0, 0) ¡Q:.
(II) A =
λ 0
1 λ
.
àg5©§| (4.2.4) )
x = 0, ½ y(x) = cx+x
λln |x|,
Ù¥ c ´?¿~ê. q
limx→0
y(x) = 0, limx→0
dy(x)
dx= limx→0
(c+
1
λln |x|+ 1
λ
)=
−∞, λ > 0,
∞, λ < 0,
¤±©§| (4.2.4) )3 (0, 0) ÛÜ(Xã 4.4. d (0, 0) ¡òz(:
(½ü(:).
ã 4.4 ½òz(:ؽòz(:
107
1oÙ 5©§ÄnØÚ)
ã 4.5 ¥%!½:ؽ:
ã 4.6 5áVòzÐÛ:
(III) A =
α −β
β α
.
ÏL4IC x = r cos θ, y = r sin θ, àg5©§| (4.2.4) z
dr
dt= αr,
dθ
dt= β.
§k)
r = c exp
(α
βθ
),
Ù¥ c ≥ 0 ´?¿~ê. ¤±©§| (4.2.4) )3 (0, 0) ÛÜ(Xã 4.5. ?
Ú/,
– α = 0 , r = c, (0, 0) ¡¥%. d (0, 0) ¿÷±Ï;.
– α > 0 , (0, 0) ¡Ø½:. d (0, 0) ; t O\Ñ^
/lmTÛ:.
– α < 0 , (0, 0) ¡½:. d (0, 0) ; t O\Ñ^/
%CTÛ:.
(IV) A =
λ 0
0 0
.
d y ¶þ:Ñ´Û:. àg5©§| (4.2.4) Ï)
x = c1eλt, y = c2,
Ù¥ c1, c2 ´?¿~ê. ¤±©§| (4.2.4) )3 (0, 0) ÛÜ(Xã 4.6. d
y ¶þ¿÷Û:, ùÛ:¡á, VòzÐÛ:.
108
§4.2 ~Xê5©§|)
ã 4.7 5ápÛ:
(V) A =
0 0
1 0
.
d y ¶þ:EÑ´Û:. àg5©§| (4.2.4) Ï)
x = c1, y = c1t+ c2,
Ù¥ c1, c2 ´?¿~ê. ¤±©§| (4.2.4) )3 (0, 0) ÛÜ(Xã 4.7. d
y ¶þ¿÷Û:, ùÛ:´á, §üAÑ", §Ñ´p
Û:.
o(þã©ÛXe(Ø.
½n46. P det(A− λE) = λ2 + pλ+ q, Ù¥ p = −trA = −(a+ d), q = det A = ad− bc. ~
Xêàg5©§| (4.2.4) :ke5.
(a) q < 0 (A küÉÒA), (0, 0) ´Q:;
(b) q = 0 (A k"A), (0, 0) ´òzÐÛ:½pÛ:;
(c) q > 0, p2 > 4q (A küØÓÒA), (0, 0) ´(:;
(d) q > 0, p2 = 4q (A küA), (0, 0) ´.(:½òz(:;
(e) q > 0, 0 < p2 < 4q (A ké¢ÜØ"A), (0, 0) ´:;
(f) q > 0, p = 0 (A kéXJA), (0, 0) ´¥%.
éu AØäkIO.,©§| (4.2.4)3:ÛÜã(JÌ´Q:Ú(:
¹. XÛÑùA«¹e@A^AÏ), ¶Ó;Úo«£ÇÖ [16] ¥k
B. Öý0XÛ$^ Mathematica 5©§|ÛÜã.
109
1oÙ 5©§ÄnØÚ)
§4.2.4 ^ Mathematica ¦§|)Ú²¡©§)ÛÜã
e¡ÏLü~f`²XÛ^ Mathematica ¦)©§|, ±9^ Mathematica
²¡©§|3Û:ÛÜã.
~K:
1. ^ Mathematica ¦©§|Ï)
x′(t) = 2x− y, y′(t) = x− 2y.
Ñ\µ
DSolve[x′[t] == 2x[t]− y[t], y′[t] == x[t]− 2 y[t], x[t], y[t], t]
Shift+Enter ÑÑ(J
x[t]→ 16e−√
3t(3− 2√
3 + 3e2√
3t + 2√
3e2√
3t)C[1]− e−√
3t(−1+e2√
3t)C[2]
2√
3,
y[t]→ e−√
3t(−1+e2√
3t)C[1])
2√
3− 1
6e−√
3t(−3− 2 3√−3e2
√3t + 2
√3e2√
3t)C[2]
2. ^ Mathematica ¦©§|ЯK)
x′(t) = 2x− 3y + e−t, y′(t) = x− 2y + e2t, x(0) = 1, y(0) = −1
Ñ\µ
DSolve[x′[t] == 2x[t]− 3 y[t] + Exp[−t], y′[t] == x[t]− 2 y[t] + Exp[2 t],
x[0] == 1, y[0] == −1, x[t], y[t], t]
Shift+Enter ÑÑ
x[t]→ − 14e−t(13− 21e2t + 4e3t + 2t), y[t]→ 1
4e−t(−11 + 7e2t − 2t)
3. ^ Mathematica Ѳ¡©§|
x′(t) = x− y, y′(t) = x+ y,
3:ã. 5µN´O, :´:.
Ñ\µ
s1 = NDSolve[x′[t] == x[t]−y[t], y′[t] == x[t]+y[t], x[0] == 0.5, y[0] == 0.5, x[t], y[t], t,−15, 15];
s2 = NDSolve[x′[t] == x[t]−y[t], y′[t] == x[t]+y[t], x[0] == 1.1, y[0] == 0.0, x[t], y[t], t,−15, 15];
s3 = NDSolve[x′[t] == x[t]−y[t], y′[t] == x[t]+y[t], x[0] == −1.1, y[0] == 1.1, x[t], y[t], t,−15, 15];
ParametricPlot[Evaluate[x[t], y[t]/.s1],Evaluate[x[t], y[t]/.s2],Evaluate[x[t], y[t]/.s3],
t,−15, 15, ImageSize→ 500,PlotRange→ −1.5, 1.5]
110
§4.3 ~Xêp5©§)
Shift+Enter =ÑÑn^;.
N5: ~ 3 ¥xÑn^;. XJF"ÏLõ;5)©§|3:
(, ±õЩ:, ±ò$m t .
~¥Jø«xÛÜã, Öö±&¢Ù§x.
§4.3 p~Xê5©§)
dup©§±=z¤§|, Ïdp~Xê5©§¦)¯K
)û. òp5©§=z¤§|3Oþ5éõØB.up~Xê
5©§gA:, I&¦¦)B.
Ä n ~Xêàg5©§
L(y) := y(n) + a1y(n−1) + . . .+ an−1y
′ + any = f(x), (4.3.1)
ÚÙéAàg5©§
y(n) + a1y(n−1) + . . .+ an−1y
′ + any = 0, (4.3.2)
Ù¥ a1, . . . , an ∈ R, f(x) 3m«m J = (a, b) þëY. e¡?ا#).
§4.3.1 ~Xêàg5©§)
ÏLò n~Xê5©§=z¤§|,±9§|A§,·±
p5©§ (4.3.1) ½ (4.3.2) A§
P (λ) = λn + a1λn−1 + . . .+ an−1λ+ an = 0. (4.3.3)
dd, p~Xê5©§A§ÃIOÒ±l§.
e¡(JÑp~Xêàg5©§Ä)|.
½n47. (4.3.3) k s pØ λ1, . . . , λs ∈ C, §ê©O´ n1, . . . , ns,
n1 + . . .+ ns = n. K¼ê|
eλ1x, xeλ1x, . . . , xn1−1eλ1x, . . . , eλsx, xeλsx, . . . , xns−1eλsx, (4.3.4)
´ n ~Xêàg5©§ (4.3.2) Ä)|.
111
1oÙ 5©§ÄnØÚ)
y: PÒBå,e¡P a0 = 1. Äky²z xkeλlx, l = 1, . . . , s, k = 0, 1, . . . , nl−1,
´àg5©§ (4.3.2) ).
du λl, l = 1, . . . , s, ´A§ (4.3.3) nl , k
dj
dλj
(n∑i=0
aiλn−i
)∣∣∣∣∣λ=λl
= 0, j = 0, 1, . . . , nl − 1,
=
n∑i=0
ai(n− i)!
(n− i− j)!λn−i−jl = 0, l = 1, . . . , s, j = 0, 1, . . . , nl − 1.
d¦È¼êê Leibniz úª
(f(x)g(x))(m) =
m∑j=0
m
j
f (j)g(m−j),
n∑i=0
ai(xkeλlx
)(n−i)=
n∑i=0
ai
n−i∑j=0
n− i
j
(xk)(j)(eλlx)(n−i−j)
=
n∑i=0
ai
k∑j=0
n− i
j
k!
(k − j)!xk−jλn−i−jl eλlx (4.3.5)
=
n∑i=0
ai
k∑j=0
k
j
(n− i)!(n− i− j)!
xk−jλn−i−jl eλlx
=
k∑j=0
k
j
xk−jeλlx
(n∑i=0
ai(n− i)!
(n− i− j)!λn−i−jl
)= 0,
Ù¥31ª¥^¯¢ (xk)(j) = 0, j > k;
n− i
j
= 0, j > n − i. ùÒy²
xkeλlx, l = 1, . . . , s, k = 0, 1, . . . , nl − 1, ´§ (4.3.2) ).
Ùgy²)| (4.3.4) 3 R þ5Ã'. P (4.3.4) ¥¼êg y1(x), . . . , yn(x).
112
§4.3 ~Xêp5©§)
K§3 R þ5Ã'=§ Wronsky 1ª
W (x) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
y1(x) y2(x) . . . yn(x)
y′1(x) y′2(x) . . . y′n(x)
......
...
y(n−2)1 (x) y
(n−2)2 (x) . . . y
(n−2)n (x)
y(n−1)1 (x) y
(n−1)2 (x) . . . y
(n−1)n (x)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣6= 0, x ∈ R.
y. d Liouville úª, b W (x) ≡ 0, x ∈ R. x0 6= 0. Kþã Wronsky 1ª¥1
þ3 x0 5'. l 3Ø"~ê b0, b1, . . . , bn−1 ¦
b0y(n−1)j (x) + b1y
(n−2)j (x) + . . .+ bn−2y
′j(x) + bn−1yj(x)
∣∣∣x=x0
= 0, j = 1, . . . , n.
du yj(x) ∈ xkeλlx; l = 1, . . . , s, k = 0, 1, . . . , nl − 1, ¤±lª (4.3.5)
n−1∑i=0
bi(xkeλlx
)(n−1−i)=
k∑j=0
k
j
xk−jeλlx
(n−1∑i=0
bi(n− 1− i)!
(n− 1− i− j)!λn−1−i−jl
).
é l = 1, . . . , s, k = 0, 1, . . . , nl − 1 k
k∑j=0
k
j
xk−jeλlx
(n−1∑i=0
bi(n− 1− i)!
(n− 1− i− j)!λn−1−i−jl
)∣∣∣∣∣∣∣x=x0
= 0. (4.3.6)
é ∀ l ∈ 1, . . . , s, 3 (4.3.6) ¥ k = 0, l j = 0, Kk
n−1∑i=0
bi(n− 1− i)!(n− 1− i)!
λn−1−il = 0.
3 (4.3.6) ¥ k = 1, (Üþª
n−1∑i=0
bi(n− 1− i)!(n− 2− i)!
λn−2−il = 0.
Uìþãg
n−1∑i=0
bi(n− 1− i)!
(n− 1− i− j)!λn−1−i−jl = 0, j = 2, . . . , nl − 1.
l é ∀ l ∈ 1, . . . , s
dj
dλj
(n−1∑i=0
biλn−1−i
)∣∣∣∣∣λ=λl
= 0, j = 0, 1, . . . , nl − 1.
113
1oÙ 5©§ÄnØÚ)
ù`² λl, l ∈ 1, . . . , s ´ê§
n−1∑i=0
biλn−1−i = 0,
nl . Ï Tê§k n1 + . . . + ns = n (Oê). gê
õ n − 1 gê§ØUk n . ùgñ`²)| (4.3.4) Wronsky 1ª
W (x) 6= 0, x ∈ R. ¤± (4.3.4) ´àg5§ (4.3.2) Ä)|. y..
N5:
1. XJkEA, 'X λl, λl, xkeλlx ½ xkeλlx ¢ÜÚJÜ=é¢).
~K:
1. ¦§
y(7) + 3y(6) + 5y(5) + 7y(4) + 7y′′′ + 5y′′ + 3y′ + y = 0,
Ï).
): A§
λ7 + 3λ6 + 5λ5 + 7λ4 + 7λ3 + 5λ2 + 3λ+ 1 = (λ+ 1)3(λ2 + 1)2 = 0,
= λ1 = −1 (n), λ2 = ±√−1 (ÝE). §kÄ)|
e−x, xe−x, x2e−x, e√−1x, xe
√−1x, e−
√−1x, xe−
√−1x.
¤±§Ï)
y(x) = c1e−x + c2xe
−x + c3x2e−x + c4 cosx+ c5x cosx+ c6 sinx+ c7x sinx,
Ù¥ c1, . . . , c7 ´?¿~ê.
2. β ∈ R ´"~ê, f(x) ´± ω ±ÏëY¼ê. y²©§
y + 2βy + y = f(x),
kk±Ï ω ±Ï).
y: ´uO§éAA§A λ1,2 = −β ±√β2 − 1.
114
§4.3 ~Xêp5©§)
i) |β| = 1. K λ = −β (). dàg5©§Ï)úª (4.1.14)
§Ï)
y(x) = c1e−βx + c2xe
−βx +
∫ x
0
(x− s)eβ(s−x)f(s)ds,
Ù¥ c1, c2 ´?¿~ê.
du
y(x+ ω) = c1e−β(x+ω) + c2(x+ ω)e−β(x+ω) +
∫ x
−ω(x− t)eβ(t−x)f(t)dt, x ∈ R,
y(x) ´±Ï ω ±Ï)¿^´
y(x+ ω) ≡ y(x), x ∈ R,
¤±éªz y(x) ´± ω ±Ï±Ï)¿^´[c1(e−βω − 1) + c2ωe
−βω −∫ 0
−ωteβtf(t)dt
]+
[c2(e−βω − 1) +
∫ 0
−ωeβtf(t)dt
]x ≡ 0, x ∈ R.
d§du
c1(e−βω − 1) + c2ωe−βω −
∫ 0
−ωteβtf(t)dt = 0,
c2(e−βω − 1) +
∫ 0
−ωeβtf(t)dt = 0.
Ï βω 6= 0, ¤±þã5ê§|k)
c1 =1
1− e−βω
∫ 0
−ω
(ω
eβω − 1− t)eβtf(t)dt, c2 =
1
1− e−βω
∫ 0
−ωeβtf(t)dt.
§k± ω ±Ï±Ï).
ii) |β| 6= 1. - % =√β2 − 1. K§Ï)
y(x) = c1e(−β+%)x + c2e
(−β−%)x +1
2%
∫ x
0
(e(−β+%)(x−s) − e(−β−%)(x−s)
)f(s)ds,
Ù¥ c1, c2 ´?¿~ê. du
y(x+ω) = c1e(−β+%)(x+ω) +c2e
(−β−%)(x+ω) +1
2%
∫ x
−ω
(e(−β+%)(x−t) − e(−β−%)(x−t)
)f(t)dt,
115
1oÙ 5©§ÄnØÚ)
¤±é y(x+ ω) ≡ y(x) z, y(x) ´± ω ±Ï±Ï)¿^´
e%x[c1(e(−β+%)ω − 1) +
1
2%
∫ 0
−ωe(β−%)tf(t)dt
]+e−%x
[c2(e(−β−%)ω − 1)− 1
2%
∫ 0
−ωe(β+%)tf(t)dt
]≡ 0, x ∈ R.
d§du
c1(e(−β+%)ω − 1) +1
2%
∫ 0
−ωe(β−%)tf(t)dt = 0,
c2(e(−β−%)ω − 1)− 1
2%
∫ 0
−ωe(β+%)tf(t)dt = 0.
§k)
c1 =1
2%(1− e(−β+%)ω)
∫ 0
−ωe(β−%)tf(t)dt, c2 =
−1
2%(1− e(−β−%)ω)
∫ 0
−ωe(β+%)tf(t)dt.
§k± ω ±Ï±Ï).
§4.3.2 ~Xêàg5©§½Xê
!?Ø©§ (4.3.1)5 f(x)äkAÏ/ª,ÏL½Xê¦
(4.3.1) )µ
1.
f(x) = Pm(x)eµx,
Ù¥ Pm(x) ´ m gõª, ©§ (4.3.1) k/X
φ∗(x) = xkQm(x)eµx,
),Ù¥ k ´ µA§ (4.3.3)ê (XJ µØ´A§ (4.3.3)
, K k = 0), Qm(x) ´½ m gõª.
2.
f(x) = (Am(x) cos(βx) +Bm(x) sin(βx))eαx,
Ù¥ Am(x), Bm(x) ´õª, maxdegAm,degBm = m, ©§ (4.3.1) k/X
φ∗(x) = xk(Cm(x) cos(βx) +Dm(x) sin(βx))eαx,
116
§4.3 ~Xêp5©§)
), Ù¥ k ´ α+√−1β A§ (4.3.3) ê (XJ α+
√−1β Ø´
A§ (4.3.3) , K k = 0), Cm(x), Dm(x) ´½ m gõª.
N5: XJ©§ (4.3.1) ¥ f(x) Øäkþ¡½/ª, §±©¤AÚ, X
f(x) = f1(x) + . . .+ fk(x),
Ù¥z fi(x) (i = 1, . . . , k) Ñäkþã½/ª, KÏL¦ L(y) = fi(x) ) φ∗i (x),
±§ (4.3.1) )
φ∗(x) = φ∗1(x) + . . .+ φ∗k(x).
~K: ¦§
y′′ + 2y′ + y = (x2 − 5)e−x + sin(2x), (4.3.7)
Ï).
): dA§ λ2 + 2λ+ 1 = 0 A λ = −1 (). l àg§Ï)
y(x) = c1e−x + c2xe
−x,
Ù¥ c1, c2 ´?¿~ê.
Äàg§
y′′ + 2y′ + y = (x2 − 5)e−x. (4.3.8)
du µ = −1 ´A, ¤±©§ (4.3.8) k/X
φ∗1(x) = x2(ax2 + bx+ c)e−x,
). òÙ\ (4.3.8) ¿z
12ax2 + 6bx+ 2c = x2 − 5.
'T§ü> x ÓgXê a = 1/12, b = 0, c = −5/2. § (4.3.8) k)
φ∗1(x) = x2
(1
12x2 − 5
2
)e−x.
Äàg§
y′′ + 2y′ + y = sin(2x). (4.3.9)
117
1oÙ 5©§ÄnØÚ)
du α+√−1β = 0 + 2
√−1 Ø´A§, ¤±©§ (4.3.9) k/X
φ∗2(x) = a cos(2x) + b sin(2x),
). òÙ\ (4.3.9) ¿z
(−3a+ 4b) cos(2x)− (4a+ 3b) sin(2x) = sin(2x).
'T§ü> sin(2x), cos(2x) Xê a = −4/25, b = −3/25. § (4.3.9) k)
φ∗2(x) = − 4
25cos(2x)− 3
25sin(2x).
nÜþãü©§), § (4.3.7) Ï)
y(x) = c1e−x + c2xe
−x + x2
(1
12x2 − 5
2
)e−x − 4
25cos(2x)− 3
25sin(2x),
Ù¥ c1, c2 ´?¿~ê.
§4.4 CXê5©§Ä:nØ
cü!Ñ~Xê5©§|Úp~Xê5©§). éCXê
5©§Ú§|%vk), !éAaAϹ\±?Ø.
§4.4.1 ±ÏXê5©§|µFloquet nØ
éuCXê5©§| (4.1.1), XJ§éAàg§|Ä)ÝÒ±
^~êC´¦§Ï).vk¦CXêàg5©§|Ä)Ý.
!ıÏXêàg5©§|
x = A(t)x, x ∈ Rn, (4.4.1)
Ù¥ A(t) ´± T > 0 ±Ï±Ïݼê.
e¡½n (¡ Floquet ½n) ´IêÆ[ Gaston Floquet (1847–1920) 3 1883
cïá, ¦Ñ±ÏXê5©§|Ä)ÝIO..
½n48. (Floquet ½n) A(t) ´± T > 0 ±Ï¢ëY±Ïݼê, Φ(t) ´ (4.4.1)
Ä)Ý. Ke(ؤá.
118
§4.4 CXê5©§Ä:nØ
(a) Ä)Ý Φ(t) ÷v
Φ(t+ T ) = Φ(t)Φ−1(0)Φ(T ), t ∈ R.
(b) 3ÛÉ!!T ±Ïݼê Q(t), 9~êÝ B (U´¢U´E
) ¦
Φ(t) = Q(t)etB.
?Ú/, 3C x = Q(t)y e, àg5©§| (4.4.1) =z~Xê5©
§|
y = By. (4.4.2)
(c) 3ÛÉ!!2T ±Ïݼê P(t), 9¢~êÝ R ¦
Φ(t) = P(t)etR.
?Ú/, 3C x = P(t)y e, àg5©§| (4.4.1) =z~Xê5©
§|
y = Ry. (4.4.3)
y: du A(t)3 RþëY,¤±§| (4.4.1)3 RþkëY¢Ä)Ý,P Φ(t).
- Ψ(t) = Φ(t+ T ), K Ψ(t) ´ (4.4.1) Ä)Ý. 3ÛÉ¢~êÝ C ¦
Φ(t+ T ) = Φ(t)C.
(a) 3þª¥- t = 0 C = Φ−1(0)Φ(T ).
(b) du C´ÛÉ,¤±l [15]½ [10, 37],3Ý B (U´¢U´E)
¦
C = eTB.
þªy²ëN¹ §6.2.
-
Q(t) = Φ(t)e−tB.
119
1oÙ 5©§ÄnØÚ)
K§´ÛÉ!ëY!T ±Ïݼê. ¯¢þ, Q(t)ëY5d Φ(t)
Ú e−tB ëY5. Q(t) ÛÉ5d Φ(t) Ú e−tB ÛÉ5,
e−tB ÛÉ5d§´ x = −Bx Ä)Ý. Q(t) ´ T ±Ï, Ï
Q(t+ T ) = Φ(t+ T )e−(t+T )B = Φ(t)Ce−TBe−tB = Φ(t)e−tB = Q(t).
C x = Q(t)y, Kk
x = Q′(t)y + Q(t)y =(Φ′(t)e−tB −Φ(t)Be−tB
)y + Q(t)y
=(A(t)Φ(t)e−tB −Q(t)B
)y + Q(t)y = A(t)Q(t)y + Q(t)(y −By)
= A(t)x + Q(t)(y −By).
d Q(t) _5, x(t) ´±ÏXê5©§| (4.4.1) )= y(t) ´
(4.4.2) ).
(c) Ï C ´ÛÉ, ¤±l [15] ½ [10, 37] 3¢Ý R ¦
C2 = e2TR.
þªy²ëN¹ §6.2. -
P(t) = Φ(t)e−tR.
K§´ÛÉ!ëY!2T ±Ïݼê. P(t) ´ 2T ±Ï, Ï
P(t+ 2T ) = Φ(t+ 2T )e−(t+2T )R = Φ(t)C2e−2TRe−tR = Φ(t)e−tR = P(t),
Ù¥^¯¢
Φ(t+ 2T ) = Φ(t+ T )C = Φ(t)C2.
C x = P(t)y, aqu (b) y² (c) y².
y..
N5: ¢Ý C ∈M ´ÛÉ, K
1. C okÝéê, =3Ý B ¦ C = eB, P B = ln C. ¢Ý C éê B U
´¢U´E.
120
§4.4 CXê5©§Ä:nØ
2. C2 ok¢Ýéê, =3¢Ý B ¦ C2 = eB.
3. C k¢Ýéê¿^´ C vkKA, ½ C Jordan IO.¥äkKA
Jordan ¬¤éÑy (ë [37]).
4. ½n 48 y²¥Ä)ÝüL« Φ(t) = Q(t)etB Ú Φ(t) = P(t)etR, §Ñ
¡Ä)Ý Φ(t) Floquet 5..
Floquet ½n«±ÏXêàg5©§|Ä)ÝA. |^ Floquet
½n±?رÏXêàg5©§3:)5. ~Xàg5©§
x′ = a(t)x,
Ù¥ a(t) ´±Ï T ëY±Ï¼ê, k")
x(t) = e∫ t0a(s)ds.
â Floquet ½n, 3 T ±Ï¼ê p(t), 9ê b ¦
x(t) = p(t)etb.
du x(t+ T ) = x(t)c = x(t)eTb, ¤±k
e∫ t+T0
a(s)ds = e∫ t0a(s)dseTb.
qdu∫ t+Tt
a(s)ds =∫ T
0a(s)ds, ±
b =1
T
∫ T
0
a(s)ds.
5¿, b T¼ê a(s) 3±Ïþ²þ. þ b = 0 , x(t) ´±Ï). þ
b < 0 , x(t) t→∞ ªu".
e¡?رÏXêàg5©§|±Ï)35. é ∀v ∈ Rn, ©§|
(4.4.1) L (τ,v) )
x(t) = Φ(t)Φ−1(τ)v.
5¿: é ∀v ∈ Rn, þª(½©§| (4.4.1) ¤k), Ïàg5©§|
)d§Ä)ÝÚЩþ(½. 5f
v −→ Φ(T + τ)Φ−1(τ)v,
121
1oÙ 5©§ÄnØÚ)
½ÂЩ v ÷X)²L±Ï T ¤3 . ùf53u: §^u
?Ø©§| (4.4.1)±Ï)35¯K.ùf¡©§| (4.4.1)ü
f. üfA¡©§| (4.4.1) A¦ê½Floquet ¦ê.
du Φ(t+ T ) = Φ(t)Φ−1(0)Φ(T ), ¤±
Φ(T + τ)Φ−1(τ) = Φ(τ)Φ−1(0)Φ(T )Φ−1(τ).
ùÒy²A¦ê´ C = eTB A.
Eê µ ¡©§| (4.4.1) Aê½Floquet ê, XJ ρ := eµT ´
A¦ê. 5¿: A¦êTk n (Oê), Floquet ê%káõ.
~K:
1. ¦e§A¦ê
x′(t) = (cos2 t)x.
): ù´±Ï π ±ÏXê5©§. §k²)
x(t) = φ(t) = e12 t+
14 sin(2t).
¤±
c := φ−1(0)φ(π) = eπ2 .
Ïd§A¦ê eπ2 .
2. ¦e§|A¦ê
x′(t) = (cos(2πt) + 1)x, y′(t) = cos(2πt)x+ y.
): ù´±Ï 1 ±ÏXê5©§|. §kÄ)Ý
Φ(t) =
et+sin(2πt)
2π 0
et+sin(2πt)
2π + et et
, 9 Φ−1(t) =
e−t−sin(2πt)
2π 0
−e−t − e−t−sin(2πt)
2π e−t
.
¤±
C := Φ−1(0)Φ(1) =
e 0
0 e
.
Ïd©§|küA¦ê e.
122
§4.4 CXê5©§Ä:nØ
e¡(JÑA¦ê±ÏXê5©§| (4.4.1) ±Ï)m'X. ±
XJvkAO`²,·` T > 0´¼ê f(t)±Ï´ T ´±Ï,= f(t+T ) ≡
f(t), t ∈ R, é?¿½ S ∈ (0, T ) Ñk f(t+ S) 6≡ f(t), t ∈ R.
½n49. éu±Ï T ±ÏXêàg5©§| (4.4.1), e¡(ؤá.
(a) λ ´©§| (4.4.1) A¦ê¿^´©§| (4.4.1) 3")
φ(t) ¦
φ(t+ T ) = λφ(t).
(b) ©§| (4.4.1) 3" 2T ±Ï)¿^´ −1 ©§| (4.4.1)
A¦ê.
(c) ©§| (4.4.1) 3" T ±Ï)¿^´ 1 ©§| (4.4.1)
A¦ê.
y: Φ(t) ´©§| (4.4.1) Ä)Ý, C = Φ−1(0)Φ(T ).
(a) 75. λ´©§| (4.4.1)A¦ê, = λ´Ý CA.P v ´ Cé
Au λ Aþ. K©§| (4.4.1) ") φ(t) = Φ(t)v ÷v
φ(t+ T ) = Φ(t+ T )v = Φ(t)Cv = Φ(t)λv = λφ(t).
¿©5. v ∈ Rn ¦ φ(t) = Φ(t)v. Ïφ(t + T ) = λφ(t), K Φ(t + T )v = λΦ(t)v.
k Cv = λv. l λ ´©§| (4.4.1) A¦ê.
(b) ¿©5. Ï −1 ´©§| (4.4.1) A¦ê, ¤±3§| (4.4.1) "
) φ(t) ¦ φ(t+ T ) = −φ(t). k φ(t+ 2T ) = −φ(t+ T ) = φ(t), = φ(t) ´ 2T ±Ï
).
75. φ(t) ´©§| (4.4.1) " 2T ±Ï), P φ(t) = Φ(t)v,
v ∈ Rn \ 0. K Φ(t+ 2T )v = Φ(t)v, l C2v = v.
du 0 = (C2 −E)v = (C + E)(C−E)v, XJ −1 Ø´ C A, K C + E
_, l (C−E)v = 0. ù`² 1´ CA. φ(t+ T ) = Φ(t+ T )v = Φ(t)Cv =
Φ(t)v = φ(t). ù φ(t) ´ 2T ±Ï)gñ. ùgñ`² −1 ´ C A, l ´
©§| (4.4.1) A¦ê.
123
1oÙ 5©§ÄnØÚ)
(c) y²aqu (b). öS3Öö. y..
N5: ±þ|^±ÏXê5©§|A¦êÑ T ±Ï)Ú 2T ±Ï)3
½. Ù¢±|^A¦ê½©§| (4.4.1)²) x = 0 ½5. k'
½5Vgò31ÊÙ¥Ñ.
e¡(JѱÏXê5©§|A¦êXêÝ A(t) ,'X.
·K50. b λ1, . . . , λn ´±Ï T ±ÏXê5©§| (4.4.1) A¦ê, K
λ1 · · ·λn = e∫ T0
tr[A(t)]dt
y: |^A¦ê½ÂÚ Liouville úª±N´y. öSÖögC¤.
~K: Hill §
x′′ + q(t)x = 0,
´ George W. Hill [28] ïÄ3±ÏÚå|^e$Ä, Ù¥ q(t) ´±Ï T
ëY±Ï¼ê. - y = x′, ±ÏXê5©§| x′
y′
=
0 1
−q(t) 0
x
y
.
|^·K 50 , T§|A¦ê¦È 1.
N5µHill §3U©!åÆ!>fó§¯õÆ¥k2A^. ©z [42] é Hill
§k~¦ïÄ.
§4.4.2 CXêàg5©§: '½nÚ Sturm-Liouville >¯K
e¡Äk?ØCXêàg5©§)":¯K.
!Sturm '½n
!ÌÄCXêàg5©§
y′′ + p(x)y′ + q(x)y = 0, (4.4.4)
)":5, Ù¥ p(x), q(x) 3m«m J = (a, b) ⊂ R þëY. T¯K3ÄïÄ
¡²~.
124
§4.4 CXê5©§Ä:nØ
3 §4.1.3 ¥®²y²µ§ (4.4.4) ?) φ(x) 3 J ?4f«mþõkk
":,3 φ(x)": x0 k φ′(x0) 6= 0. e¡?Ú?Ø©§ (4.4.4))":5
.
¡ x1, x2 ∈ J ´¼ê φ(x) ü":, XJ x1, x2 Ñ´ φ(x) ":, 3 x1
x2 mvk φ(x) Ù§":.
ÄkÏLü~fwàg5©§ü)":m'X. à
g5©§
y′′ + ω2y = 0, ω > 0 ´?¿½~ê,
Ï)
y = c1 cos(ωx) + c2 sin(ωx),
Ù¥ c1, c2 ´?¿~ê. §ü5Ã')
y1(x) = cos(ωx), y2(x) = sin(ωx),
":3 R þü. ±y, Ï)¥?ü5Ã')Ñäkù«5; ?ü
5')ÑäkÓ":.
þã~f¥ü)":5´àg5©¤äkA5.
·K51. y = φ1(x) Ú y = φ2(x) ´àg5©§ (4.4.4) ü"), Ñ
k":. Ke(ؤá.
(a) y = φ1(x) y = φ2(x) 3 J þ5Ã'=§":p.
(b) y = φ1(x) y = φ2(x) 3 J þ5'=§kÓ":.
y: P W (x) φ1(x), φ2(x) Wronsky 1ª.
(a) 75. du φ1(x), φ2(x) 3 J þ5Ã', ¤± W (x) 6= 0, x ∈ J . db,
φ1(x), φ2(x) 3 J þÑk":.
XJ φ1(x), φ2(x) kÓ":, P x0 ∈ J . K
W (x) = W (x0)e−
∫ xx0p(s)ds
= 0, x ∈ J,
bgñ. φ1(x), φ2(x) 3 J þvkÓ":.
125
1oÙ 5©§ÄnØÚ)
XJ φ1(x) φ2(x) Ñk":, Ïùü":ØÓ, (Øw,¤á. Øb
φ1(x) kü":, x1, x2 ´ φ1(x) ü":. 2Ø x1 < x2
φ1(x) > 0, x ∈ (x1, x2).
K
φ′1(x1) > 0, φ′1(x2) < 0.
Ï W (x) 6= 0, ¤± W (x1)W (x2) > 0. q
W (x1) = −φ2(x1)φ′1(x1), W (x2) = −φ2(x2)φ′1(x2),
k
φ2(x1)φ2(x2) < 0.
d φ2(x) ëY5: φ2(x) 3 (x1, x2) þ7k":k":. ÄKÓþy²,
φ1(x) 3 φ2(x) u (x1, x2) þ":mk":, x1, x2 ´ φ1(x) ":gñ. ù
Òy² φ1(x) φ2(x) ":p.
¿©5. y. XJ φ1(x) φ2(x) 5', K3Ø"~ê c1, c2 ¦
c1φ1(x) + c2φ2(x) ≡ 0, x ∈ J.
¯¢þ c1, c2 ÑØ", ÄK φ1(x) φ2(x) ¥7kð", ù φ1(x) Ú φ2(x) Ñ´
")bgñ. l
φ2(x) = cφ1(x), c 6= 0, x ∈ J.
ù`² φ1(x) φ2(x) ":Ó, bgñ.
(b) d (a) y²N´. öSdÖögC¤. y..
e¡0àg5©§¥~(صSturm'½n,§Ñü
àg5©§)":m'X.
½n52. (Sturm '½n) Äàg5©§
y′′ + p(x)y′ + q(x)y = 0, (4.4.5)
y′′ + p(x)y′ + r(x)y = 0, (4.4.6)
Ù¥ p(x), q(x), r(x) 3m«m J þëY. y = φ(x) Ú y = ψ(x) ©O´§ (4.4.5) Ú
(4.4.6) "), φ(x) kü": x1, x2 ∈ J . Ø x1 < x2. Ke(ؤá.
126
§4.4 CXê5©§Ä:nØ
(a) XJ r(x) ≥ q(x), x ∈ J , K ψ(x) 3 [x1, x2] þk":.
(b) XJ r(x) ≥ q(x), x ∈ J , q(x) 6≡ r(x), x ∈ (x1, x2), K ψ(x) 3 (x1, x2) þk
":.
y: (a) Ø φ(x) > 0, x ∈ (x1, x2). K
φ′(x1) > 0, φ′(x2) < 0.
y, b ψ(x) 6= 0, x ∈ [x1, x2]. Ø
ψ(x) > 0, x ∈ [x1, x2].
-
V (x) = ψ(x)φ′(x)− φ(x)ψ′(x).
K
V (x2) = ψ(x2)φ′(x2) < 0, V (x1) = ψ(x1)φ′(x1) > 0. (4.4.7)
qO
V ′(x) + p(x)V (x) = ψ(x)φ′′(x)− φ(x)ψ′′(x) + p(x)V (x)
= (r(x)− q(x))φ(x)ψ(x) ≥ 0, x ∈ [x1, x2].
ü>¦± e∫ xx1p(s)ds
, ¿l x1 x2 È©
V (x2)e∫ x2x1
p(s)ds ≥ V (x1).
ù (4.4.7) gñ. ¤± ψ(x) 3 [x1, x2] þk":.
(b). |^y, l (a) y²±9∫ x2
x1
(r(x)− q(x))φ(x)ψ(x)e∫ xx1p(s)ds
dx > 0,
±y. y..
|^ Sturm '½n±½/X (4.4.5) ©§")":35.
íØ53. p(x), q(x) ∈ C([a,∞)), a ∈ R, y = φ(x) ´§ (4.4.5) "), Ke
(ؤá.
127
1oÙ 5©§ÄnØÚ)
(a) XJ q(x) ≤ 0, x ∈ [a,∞), K φ(x) 3 [a,∞) þõk":.
(b) XJ p(x) ≡ 0, q(x) ≥ m > 0, x ∈ [a,∞), K φ(x) 3 [a,∞) þkáõ":,
":mål ≤ π√m
.
(c) XJ p(x) ≡ 0, q(x) > m > 0, x ∈ [a,∞), K φ(x) 3 [a,∞) þkáõ":,
":mål < π√m
.
(d) XJ p(x) ≡ 0, 0 < q(x) ≤ m, x ∈ [a,∞), K φ(x) ":mål ≥ π√m
.
(e) XJ p(x) ≡ 0, 0 < q(x) < m, x ∈ [a,∞), K φ(x) ":mål > π√m
.
y: íØy²ÌÏL·'§, |^ Sturm '½n¤.
(a) 3§ (4.4.6) ¥ r(x) ≡ 0, K ψ(x) ≡ 1 ´Ù). d Sturm '½n, XJ φ(x)
kü":, K ψ(x) 3ùü":m7k":, gñ. ¤± φ(x) kõ"
:.
(b) Ä (4.4.5) '§
y′′ +my = 0. (4.4.8)
§k) y = ψ(x) = sin (√m(x− a)) , x ∈ [a,∞). w, ψ(x) káõ":
xn = a+nπ√m, n = 0, 1, . . .
â Sturm '½n (a), φ(x) 3z«m[a+
nπ√m, a+
(n+ 1)π√m
], n = 0, 1, . . . ,
þÑk":. § φ(x) 3 [a,∞) þkáõ":.
x∗1, x∗2 ´ φ(x) 3 [a,∞) þü":, Ø x∗1 < x∗2. y, b
x∗2 − x∗1 >π√m.
éu?¿ b ∈ R, du
sin(√m(x− a− b)
), x ∈ [a,∞),
´§ (4.4.8) ), §":mål´π√m
. ¤±· b
¦ sin (√m(x− a− b)) kü": u (x∗1, x
∗2) SÜ, P x1, x2, ÷v
128
§4.4 CXê5©§Ä:nØ
x∗1 < x1 < x2 < x∗2. 2|^ Sturm '½n (a) , φ(x) 3 [x1, x2] þk":. ù
x∗1, x∗2 ´ φ(x) ":gñ. ùgñ`² x∗2 − x∗1 ≤
π√m
.
(c) d (a) y², § (4.4.5) ) φ(x) 3 [a,∞) þkáõ":, Ù":
målÑ ≤ π√m
.
y,XJ φ(x)kü": x∗1, x∗2 ÷v x∗2−x∗1 =
π√m
. aqu (b)y²,
· c ¦ sin (√m(x− a− c)) kü":T u x∗1, x
∗2. Kd Sturm
'½n (b) , φ(x) 3 (x∗1, x∗2) SÜk":. ù x∗1, x
∗2 ´ φ(x) ":g
ñ. ¤± 0 < x∗2 − x∗1 <π√m
.
(d), (e) §y²aqu (b) Ú (c)y². öSÖögC¤.
y..
§ (4.4.5) )XJ3 J þkáõ":, K¡§3 J þ´½Ä.
N5:
• 3íØ 53 (b) Ú (c) ¥, XJk q(x) > 0, (Ø7¤á. ~X§
y′′ +a
4x2y = 0, a > 0 ´?¿½~ê,
Ï)
y(x) =
x
12 (c1 + c2 lnx), a = 1,
c1x1+√
1−a2 + c2x
1−√
1−a2 , a < 1,
x12
(c1 cos(
√a−12 lnx) + c2 sin(
√a−12 lnx)
), a > 1.
a ≤ 1 , ?")3 [1,∞) þõk":; a > 1 , ?")3
[1,∞) þkáõ":.
!Sturm-Liouville >¯K
!ü/?Øàg5©§>¯K.T¯K3¢S)¹¥²~, X
²¡þüà½.lüà:¤3tm,¯$Ä5ÆXÛ?üà
½g^3Øå^e5XÛ? ùÑ´©§>¯KïÄSN.
Ä Sturm-Liouville ©§
(p(x)y′)′ + (λr(x) + q(x))y = 0, λ ∈ R ëê, (4.4.9)
129
1oÙ 5©§ÄnØÚ)
Ù¥ q, r ∈ C[a, b], p ∈ C1[a, b], r(x) > 0. ±eÑ3Tb^e?Ø. PÒBå,
-
Ly = (p(x)y′)′ + q(x)y.
Xe>¯K
Ly = −λr(x)y,
αy(a)− βy′(a) = 0, γy(a) + δy(b) = 0,(4.4.10)
Ù¥ α, β, γ, δ ∈ R, α2 + β2 6= 0, γ2 + δ2 6= 0, ¡Sturm-Liouville >¯K, ¡ SL >
¯K. 3 a, b ü:^¡>.^.
XJé λ = λ0, >¯K (4.4.10) k") y0(x), K¡ λ0 ´ SL >¯KA
, ¡ y0(x) ´ SL >¯KAu λ0 A¼ê.
~K: y SL >¯K
y′′ = −λy,
y(0) = 0, y(π) = 0,
kAÚAA¼ê
λn = n2, yn(x) = sin(nx), n = 1, 2, . . .
): \>¯Ky=.¯¢þ,TAÚA¼ê±ÏL¦§ y′′ = −λy
Ï)¿|^>.^.
þ~¥A÷v λ1 < λ2 < . . . < λn < . . ., limn→∞
λn = ∞, AuØÓA
A¼êüü, = ∫ π
0
sin(mx) sin(nx)dx = 0, m, n ∈ N, m 6= n.
?Ú/ sin(nx) 3 (0, π) þTk n− 1 ":. e¡òy² SL >¯K÷vù
5.
?Ú?Ø SL>¯K)5,ÄkÑ#½Â.éu½ r(x) ∈ C[a, b]
÷v r(x) ≥ 0, r(x) 6≡ 0.
• ¼ê f(x), g(x) ∈ C[a, b] 'u r(x) \SȽÂ
〈f, y〉r =
∫ b
a
r(x)f(x)g(x)dx.
130
§4.4 CXê5©§Ä:nØ
• ¼ê r(x) ¡\¼ê.
• XJ f, g 'u r \SÈ", ¡ f, g 'u\¼ê.
5µXJ r = 1, @o\SÈÒ´Ï~üXþ¼êSÈ.
½n54. éu SL >¯K, b p ∈ C1[a, b], q, r ∈ C[a, b], r(x) > 0, α, β, γ, δ ∈ R,
α2 + β2 6= 0, γ2 + δ2 6= 0. e(ؤá.
(a) SL >¯KAe3Ñ´¢;
(b) AuzA, SL >¯Kk¢A¼ê;
(c) AuzA, SL >¯Kk5Ã'A¼ê;
(d) AuØÓAA¼ê'u\¼ê r(x) ´;
(e) AuØÓAA¼ê5Ã'.
yµ λ1, λ2 ´ SL >¯KA, y1(x), y2(x) ´AA¼ê. KdAÚA
¼ê½Â, ÏLO
y1Ly2 − y2Ly1 = p′(x)(y1(x)y′2(x)− y2(x)y′1(x))
+p(x)(y1(x)y′′2 (x)− y2(x)y′′1 (x)) = (p(x)W (x))′,
Ù¥ W (x) ´ y1, y2 Wronsky 1ª. Ïdk
(λ1 − λ2)r(x)y1(x)y2(x) = (p(x)W (x))′.
éþªl a b È©
(λ1 − λ2)〈y1, y2〉r = 0, (4.4.11)
þª¥^ W (a) = W (b) = 0. ¯¢þ, Ï
αy1(a)− βy′1(a) = 0, αy2(a)− βy′2(a) = 0,
q α2 + β2 6= 0, Ø α 6= 0, ¤±l1§¦± y′2(a) 1§¦± y′1(a)
= W (a) = 0. Ónlb :>.^= W (b) = 0.
131
1oÙ 5©§ÄnØÚ)
(a) du SL >¯KXêÑ´¢, XJ§kEAÚA¼ê, @oùAÚ
A¼êÝ´ SL >¯KAÚA¼ê.
- λ0 Ú y0(x) ´ SL >¯KAÚA¼ê. Kk
(λ0 − λ0)〈y0, y0〉r = 0.
Ï 〈y0, y1〉r =∫ bar(x)|y0(x)|2dx 6= 0, ¤± λ0 − λ0 = 0. l λ0 ´¢ê.
(b) λ0 Ú y0(x) ´ SL >¯KAÚA¼ê. d (a), λ0 ´¢ê. XJ y0(x) ´
¢¼ê, K (b) ¤á. XJ y0(x) = u(x) +√−1v(x), KÏ>¯KXêÚ λ0 Ñ´¢
, l u(x) Ú v(x) Ñ´Au λ0 A¼ê.
(c) λ0´ SL>¯KA, y1(x)Ú y2(x)´Au λ0A¼ê. du y1(x), y2(x)
Ñ´àg5§ (4.4.9) λ = λ0 ), § Wronsky 1ªÑ3 a :u",
¤± y1(x) y2(x) 5'.
(d) y1(x), y2(x)´©OAuØÓA λ1, λ2 A¼ê. Kd (4.4.11) 〈y1, y2〉r =
0. Ïd y1, y2 'u\¼ê r(x) ´.
(e) y1(x), y2(x) ´AuØÓAA¼ê, k~ê c1, c2 ¦
c1y1(x) + c2y2(x) ≡ 0, x ∈ [a, b].
§ü>Ó¦± r(x)y2(x), ¿l a b È© c2∫ bar(x)y2
2(x)ds = 0. l c2 = 0. Ó
ny c1 = 0. ùÒy² y1, y2 3 [a, b] þ5Ã'.
½ny..
e¡(JÑ SL >¯KáõA35, ±9A¼ê":5.
½n55. éu SL >¯K, b p ∈ C1[a, b], q, r ∈ C[a, b], r(x) > 0, α, β, γ, δ ∈ R,
α2 + β2 6= 0, γ2 + δ2 6= 0. e(ؤá.
(a) SL >¯KkáõA λi, i ∈ N ÷v
λ1 < λ2 < . . . < λn < . . . , limn→∞
λn =∞;
(b) yn(x) ´Au λn A¼ê, K yn(x) 3 (a, b) þTk n− 1 ":.
y: T½ny²E,, ÖlÑ. k,Ööë[16, 51].
132
§4.4 CXê5©§Ä:nØ
§4.4.3 pCXê5©§µ?ê)
pCXêàg5©§Ä)|vk¦. !ÄAÏ
àg5©§
y′′ + p(x)y′ + q(x)y = 0, (4.4.12)
?ê).
XJ p(x), q(x) 3 x0 ,)Û, ¡ x0 (4.4.12) ~:. XJ p(x) ½ q(x) 3 x0
Ø)Û, ¡ x0 (4.4.12) Û:.
Äkħ (4.4.12) 3~:)Û)Âñ».
½n56. p(x), q(x) 3 |x−x0| < ρ SФ'u x−x0 Âñ?ê, K§ (4.4.12)
3 |x− x0| < ρ SkÂñ?ê)
y(x) =
∞∑k=0
ck(x− x0)k,
Ù¥ c0, c1 ´?¿~ê, ck, k > 1 d4íúªÏL c0, c1 L«. AO/, c0, c1 dЩ^
(½.
y: duàg5©§ (4.4.12) ÷vЩ^
y(x0) = y0, y′(x0) = y1, y0, y1 ∈ R, (4.4.13)
ЯKÏLC y′ = z zàg5©§|
y′ = z, y(x0) = y0,
z′ = −p(x)z − q(x)y, z(x0) = y1,
Ïd½ny²d½n 27 . y..
ÙgÄàg5©§ (4.4.12) 3Û:)Û)35. XJ
p(x) =P (x)
x− x0, q(x) =
Q(x)
(x− x0)2, (4.4.14)
P (x), Q(x) 3 x0 ,SФÂñ?ê, P (x0)2 + Q(x0)2 6= 0, ¡ x0 ´§
(4.4.12) KÛ:.
133
1oÙ 5©§ÄnØÚ)
½n57. x0 ´§ (4.4.12) KÛ:, p(x), q(x) d (4.4.14) ½Â. K§ (4.4.12)
3 x0 ,SkÂñ2Â?ê)
y(x) = (x− x0)ν∞∑k=0
ck(x− x0)k, c0 6= 0, (4.4.15)
Ù¥ ck, k ≥ 1 ±S/¦Ñ, ν ´§ (4.4.12) I§
s(s− 1) + P (x0)s+Q(x0) = 0,
(¡I) (XJIÑ´¢, ν ´Ù¥; XJI´é
ÝEê, ν ´Ù¥?).
y: 1. (½§ (4.4.12) /ª) (4.4.15).
dKÛ:½Â, Ø P (x), Q(x) 3 |x− x0| < ρ SФÂñ?ê
P (x) =
∞∑k=0
ak(x− x0)k, Q(x) =
∞∑k=0
bk(x− x0)k. (4.4.16)
ò (4.4.15) Ú (4.4.16) \§ (4.4.12), ÏLn
∞∑k=0
[(k + ν)(k + ν − 1)ck +
k∑i=0
ai(k − i+ ν)ck−i +
k∑i=0
bick−i
](x− x0)k ≡ 0,
=
[(k + ν)(k + ν − 1) + a0(k + ν) + b0] ck +
k∑i=1
[ai(k + ν − i) + bi] ck−i = 0, k = 0, 1, . . .
(4.4.17)
Ù¥ cj = 0, j < 0.
lþª k = 0 9 c0 6= 0
ν(ν − 1) + a0ν + b0 = 0.
§´§ (4.4.12) I§. ν1, ν2 ´§ü. XJ ν1, ν2 Ñ´¢ê ν1 ≥ ν2, ½
ν1, ν2 ´éÝE, P ν0 = ν1.
é k > 0, -
f(s) = s(s− 1) + a0s+ b0, gi(s) = ai(s− i) + bi, i = 1, 2, . . .
¿ ν = ν0, K (4.4.17) ¤
f(k + ν0)ck +
k∑i=1
gi(k + ν0)ck−i = 0, k = 1, 2, . . . (4.4.18)
134
§4.4 CXê5©§Ä:nØ
Ïé k ≥ 1
f(k + ν0) = (k + ν0)(k + ν0 − 1) + a0(k + ν0) + b0
= k(k + 2ν0 + a0 − 1) = k(k + 2ν0 − ν1 − ν2) = k(k + ν1 − ν2) 6= 0,
¤±l§ (4.4.18) ±g/¦Ñ c1, c2, . . ., §Ñd c0 /(½. l /ªþ
/¦Ñ§ (4.4.12) /ª) (4.4.15).
2. y²/ª) (4.4.15) Âñ5.
- ν∗ = ν1 − ν2. K Re ν∗ ≥ 0
f(k + ν0) = k(k + ν∗) 6= 0, k = 1, 2, . . .
du P (x)Ú Q(x)Ðmª (4.4.16)3 |x−x0| < ρSÂñ,¤±é 0 < ρ1 < ρ, ∃M ≥ 1¦
|ak| ≤M
ρk1, |bk| ≤
M
ρk1, |ν0ak + bk| ≤
M
ρk1, k = 0, 1, . . .
l
|ck| ≤(M
ρ1
)k|c0|. (4.4.19)
¯¢þ, é k = 1
|c1| =|g1(1 + ν0)c0||f(1 + ν0)|
=|a1ν0 + b1||1 + ν∗|
|c0| ≤M
ρ1
1
|1 + ν∗||c0| ≤
M
ρ1|c0|.
bé k ≤ l − 1, l ≥ 2, (4.4.19) ¤á. k = l
|cl| =
∣∣∣∣ l∑i=1
gi(l + ν0)cl−i
∣∣∣∣|l(l + ν∗)|
≤|(alν0 + bl)c0|+
l−1∑i=1
|ai(l + ν0 − i) + bi| |cl−i|
|l(l + ν∗)|
≤|(alν0 + bl)|+
l−1∑j=1
|al−j(j + ν0) + bl−j |(Mρ1
)j|l|2
|c0|
≤ l−2l−1∑j=0
(|ν0al−j + bl−j |+ |jal−j |)(M
ρ1
)j|c0|
≤ l−2l−1∑j=0
(M
ρl−j1
+ jM
ρl−j1
)(M
ρ1
)j|c0|
= l−2M
ρl1
l−1∑j=0
M j(1 + j)|c0| ≤ l−2Ml
ρl1
l∑j=1
j|c0| ≤(M
ρ1
)l|c0|.
135
1oÙ 5©§ÄnØÚ)
¤±d8By (4.4.19).
l (4.4.19) ª, ©§ (4.4.12) /ª) (4.4.15) 3 |x− x0| < ρ1M SÂñ. y..
~K:
1. ¦ Legendre §
(1− x2)y′′ − 2xy′ + n(n+ 1)y = 0,
3~: x = 0 S?ê).
): â½n 56, |x| < 1 , Legendre §kÂñ?ê)
y(x) =
∞∑k=0
ckxk.
òÙ\ Legendre §, ÏLn
∞∑k=0
[(k + 2)(k + 1)ck+2 + (n− k)(n+ k + 1)ck]xk ≡ 0, |x| < 1.
k
ck+2 = − (n− k)(n+ k + 1)
(k + 2)(k + 1)ck, k = 0, 1, . . .
?ÚO
c2m = (−1)mAmc0, c2m+1 = (−1)mBmc1, m = 1, 2, . . .
Ù¥
Am =(n− 2m+ 2)(n− 2m+ 4) . . . (n− 2)n(n+ 1)(n+ 3) . . . (n+ 2m− 1)
(2m)!,
Bm =(n− 2m+ 1)(n− 2m+ 3) . . . (n− 3)(n− 1)(n+ 2)(n+ 4) . . . (n+ 2m)
(2m+ 1)!.
Ïd Legendre §?ê/ªÏ)
y(x) = c0
(1 +
∞∑m=1
(−1)mAmx2m
)+ c1x
(1 +
∞∑m=1
(−1)mBmx2m
), |x| < 1,
Ù¥ c0, c1 ´?¿~ê.
N5:
1) n ∈ N∪0,3 m0 ∈ N¦ m > m0 k Am = 0½ Bm = 0. d Legendre
§kõª), ¡ Legendre õª.
136
§4.4 CXê5©§Ä:nØ
2) x = ±1 ´ Legendre §KÛ:, §3T:IÑ´". Ïdd½n 57 ,
Legendre §3 x = ±1 ÑkÂñ?ê).
2. ¦ µ Bessel §
x2y′′ + xy′ + (x2 − µ2)y = 0, µ ≥ 0,
2µ 6∈ Z 3 x = 0 Ï).
): ´ x = 0 ´ Bessel §KÛ:, I§ ν2 − µ2 = 0. â½n 57, Bessel
§3 x = 0 kÂñ2Â?ê),
y(x) = xν∞∑k=0
ck xk, c0 6= 0.
òÙ\ Bessel §, ÏLn¿' x ÓgXê
(k + ν + µ)(k + ν − µ)ck + ck−2 = 0, k = 0, 1, . . . (4.4.20)
Ù¥ c−2 = c−1 = 0.
ν = µ , du (k + ν + µ)(k + ν − µ) > 0, k ≥ 1, ¤±l§ (4.4.20) )
c2k−1 = 0, c2k =(−1)k
22k(µ+ k)(µ+ k − 1) . . . (µ+ 2)(µ+ 1) k!c0, k = 1, 2, . . .
-
Γ(s) =
∫ ∞0
e−xxs−1dx, ¡ Gamma ¼ê.
K Γ(s+ 1) = sΓ(s). 5¿,
Γ(µ+ k + 1) = (µ+ k) . . . (µ+ 2)(µ+ 1)Γ(µ+ 1),
Γ(−s) =∞, s ≥ 0.
PÒBå,
c0 =1
2µΓ(µ+ 1).
K
c2k =(−1)k
22k+µΓ(µ+ k + 1) k!, k = 1, 2, . . .
137
1oÙ 5©§nØÚ)
Ïd Bessel §k2Â?ê)
y(x) = Jµ(x) =
∞∑k=0
(−1)k
Γ(k + 1 + µ) k!
(x2
)2k+µ
.
N´yT?ê3 |x| < ∞ þÂñ (X D’Alembert O). ¼ê Jµ(x) ¡ µ
1a Bessel ¼ê.
ν = −µ , du 2µ 6∈ Z, § (4.4.20) k).
c0 =1
2−µΓ(−µ+ 1).
aqu ν = µ ¦ Bessel §2Â?ê)
y(x) = J−µ(x) =
∞∑k=0
(−1)k
Γ(k + 1− µ) k!
(x2
)2k−µ.
§3 x ∈ R \ 0 þÂñ. ¼ê J−µ(x) ¡ µ 1a Bessel ¼ê.
du Jµ(x)Ú J−µ(x)$gØÓ,Ï §´5Ã'. ¤± Bessel
§Ï)
y(x) = c1Jµ(x) + c2J−µ(x), x ∈ R \ 0,
Ù¥ c1, c2 ´?¿~ê.
N5:
1) ~ 2 ¥?Ø 2µ 6∈ Z ¹. 2µ ∈ Z /E,, Ïd k − 2µ ±",
Ï S§ (4.4.20) æ). 3öSÖögCg.
2) þãüCXꩧ/ªþ~ü,§¦)½)LªéE,. ù
±wÑ=¦´CXê5©§Uæ)©
3) þãü~f¥§)Ñشмê,¡Aϼê. Aϼêäk~´L
SN, §3ó§þk2A^. k,Öö±ëÆM!H;Aϼê
VØ [54].
138
§4.5 SKo
§4.5 SKo
1. b b(x) > 0, x ∈ I = [c, d]. XJ φ(x), a(x), b(x) ∈ C(I), ÷vت
φ(x) ≤ a(x) +
∫ x
c
b(s)φ(s)ds,
K
φ(x) ≤ a(x) +
∫ x
c
a(s)b(s)e∫ xsb(t)dtds.
2. Á|^ Gronwall تy²½n 18 ¥)5.
3. 3½n 36 bÚPÒe, y²
y1(x) + y∗(x), . . . , yn(x) + y∗(x), y∗(x),
´àg5©§| (4.1.1) n+ 15Ã').?Ú/ (4.1.1)õk n+ 1
5Ã').
4. y²¼ê| 1
sinx
,
0
x2
, x ∈ R
ØU´?Ûàg5©§|)|.
5. y²íØ 38.
6. éuàg5©§| (4.1.2), ¡ y′ = −AT (x)y (4.1.2) ݧ, Ù¥ T L
«Ý=. Áye(Ø
(a) XJ Φ(x)´ (4.1.2)Ä)Ý,K (ΦT (x))−1 ´ (4.1.2)ݧÄ)Ý.
(b) XJΦ(x), Ψ(x)©O´ (4.1.2)9Ùݧ|Ä)Ý,K§¦ÈΦ(x)Ψ(x)
´ÛÉ~êÝ.
(c) XJ φ(x), ψ(x)©O´ (4.1.2)9Ùݧ|),K§¦È φ(x)ψ(x)´
~ê.
7. y²½n 39 N5 2 Ú 3.
139
1oÙ 5©§nØÚ)
8. ¦e§|Ï)
8.1. x′(t) =2
tx− 1, y′(t) = − 1
t3x+
1
ty +
1
t2, x(1) = 1, y(1) = 1.
8.2. x′(t) = −1
tx+ y, y′(t) = − 2
t2x+
1
ty, x(1) = 1, y(1) = 2.
9. XJpàg5©§"), KT§±ü$.
10. òeCXê5©§z¤~Xê5©§
10.1. x2y′′ + 3xy′ − 2y = 0.
10.2. x3y′′′ − 5x2y′′ + 2xy′ + 7y = 0.
11. -M(C) ´ n EÝN¤8Ü. é?¿ A,B ∈ M(C), ½Â§
f [A, B] := AB−BA. y²µXJ [A, B] = E (E ´ n ü Ý), K
etAetB = etBetAet2[A,B], t ∈ R.
J«µ1Úy²
[A, Bn] = nBn−1, [Bn, A] = nBn−1, n ∈ N \ 0.
1Ú|^þã'XO [etA, B]Ú [A, etB]. éݼêX(t) = etAetBe−tAe−tB
÷v§, ¿¦Ù÷vЩ^ X(0) = E ).
12. ^Ýê¼ê¦)e~Xê5©§
dy
dx= Ay + f(x),
Ù¥
12.1 A =
2 0
1 2
, f(x) =
−1
x
;
12.2. A =
−1 0 0
1 −1 0
0 1 −1
, f(x) =
−1
x
x
13. y²~Xêàg5©§| (4.2.2) XêÝ A A¢ÜÑu"¿
^´T§|¤k) x→∞ Ѫu".
140
§4.5 SKo
14. Φ(x) ´ R þëY n Ý, detΦ(0) 6= 0, Φ(0) 3, ÷v
Φ(t+ s) = Φ(t)Φ(s), ∀ t, s ∈ R.
Áy² Φ(x) ´, n ~Xêàg5©§Ä)Ý.
15. ¦)e~Xê5©§|Ï)
15.1. x = 3x− y, y = 2x+ y;
15.2. x = y, y = 4x+ 2y − 4z, z = x− y + z;
15.3. x = −n2y + cosnt, y = −n2x+ sinnt;
15.4. x = 4x+ y − 2, y = −2x+ y + e3t;
15.5. x = −2x+ y − z + et, y = x+ z, z = 3x− y + 2z − et;
15.6. x = x+ y − t2, y = y + z − 2t, z = z − t.
16. ¦)e~Xê5©§|ЯK
16.1. x = x+ 5y, y = −x− 3y, x(0) = 1, y(0) = 2;
16.2. x = x+ 2y + z, y = x− y + z, z = 2x+ z, x(0) = 1, y(0) = 0, z(0) = −1;
16.3. x = 4x− 4y + 2z, y = x, z = −x+ 2y − z, x(0) = 1, y(0) = 0, z(0) = 0;
16.4. x = 3x− 2y − et, y = 2− y + et, x(0) = 0, y(0) = −1;
16.5. x = x+ 1 + 12e
2, y = −2y + 12e
2, z = −2z + 2− 12e
2, x(0) = 0, y(0) = 1, z(0) = 0;
16.6. x = 2x−y+z+2, y = x+z+1, z = −3x+y−2z−3, x(0) = −1, y(0) = 1, z(0) = 2.
17. ½e²¡5©§|Û:5, ¿xÑÛÜã
17.1. x = 3x− y, y = 2x− 3y;
17.2. x = x− y, y = 2x− y;
17.3. x = −y, y = 2x− 2y;
17.4. x = x− y, y = 3x− 2y;
17.5. x = x− y, y = −2y;
141
1oÙ 5©§nØÚ)
17.6. x = −x+ y, y = −3x− 5y;
17.7. x = x+ 2y, y = −x+ y;
17.8. x = x− 2y, y = −x+ y;
17.9. x = 2x+ 2y, y = x+ 2y;
17.10. x = −2x− 3y, y = x− 2y;
17.11. x = −2y, y = x;
17.12. x = 3x+ 2y, y = 3y;
17.13. x = 2x+ 2y, y = x+ y;
17.14. x = x− y, y = x+ 3y.
18. a, b ∈ R ´~ê. éu5©§
y′′ + ay′ + by = 0, (4.5.1)
Á¯
(a) a, b Û, § (4.5.1) ¤k)3 R þk.;
(b) a, b Û, § (4.5.1) ¤k) x→∞ Ѫu";
(c) a, b Û, § (4.5.1) ¤k)3 R þÑkáõ":.
19. f(x) 3 [0,∞) þëY. éu5©§
y′′ + 4y′ + 3y = f(x), (4.5.2)
Áy
(a) XJ f(x) 3 [0,∞) þk., § (4.5.2) ¤k)3 [0,∞) þk.;
(b) XJ limx→∞
f(x) = 0, § (4.5.1) ¤k) x→∞ Ѫu".
20. ¦)e~Xê5©§
20.1. x(4) − x = 0;
20.2. x(6) − 4x(5) + 4x(4) = 0;
142
§4.5 SKo
20.3. x′′′ − 3x′′ − x′ + 3x = 0;
20.4. x′′ + 2x′ + x = 0, x(0) = 1, x′(0) = −1;
20.5. x(4) + 8x′′ + 16x = 0, x(0) = 1, x′(0) = 2, x′′(0) = 0, x′′′(0) = 0;
20.6. x′′′ − 2x′′ + x′ − 2x = 0, x(0) = −1, x′(0) = 0, x′′(0) = 0;
20.7. x′′′ + x′ = 1 + 2 cos(2t);
20.8. x′′ + x′ = t− e−t;
20.9. x(4) + 2x′′ + x = e−t;
20.10. x′′ + x = t cos t;
20.11. x′′ − x = 2et, x(0) = 0, x′(0) = 3;
20.12. x′′′ − 3x′′ + x′ − 3x = 2 sin(3t), x(0) = −1, x′(0) = 2;
20.13. x′′′ − 2x′′ = t, x(0) = −1, x′(0) = 0, x′′(0) = 1,
20.14. x′′ − 2x′ + x = tet, x(0) = 3, x′(0) = 1.
20.15. t2x′′ − 2tx′ + 2x = 0, x(1) = 1, x′(1) = 1.
21. y²½n 49 (c).
22. éuàg5±Ï©§|
x(t) = A(t)x + f(t), (4.5.3)
Ù¥ A(t) Ú f(t) ©O´ R þëY T ±ÏÝÚþ¼ê. b Φ(t) ´ (4.5.3) é
Aàg5©§|Ä)Ý, Φ(0) = E. Áy²
(a) b x = φ(t) ´ (4.5.3) ), K x = φ(t) ± T ±Ï¿^´ φ(0) = φ(T ).
(b) §| (4.5.3) 3±Ï T ±Ï)¿^´ Φ(T ) ر 1 A
.
23. b a(t), b(t) ´±Ï T ëY±Ï¼ê. ÁÑeàg5©§|
x = a(t)x, y = b(t)y,
FloquetIO.. ¿ÑäNCòT§|z¤~Xêàg5©§|. ?
Ú?ØÛk T ±Ï), Û¤k) t→∞ Ѫu".
143
1oÙ 5©§nØÚ)
24. ¦e§½§|A¦ê, ¿±d½ù§½§|´Äk T ½ 2T ±Ï
) (T Le§±Ï).
24.1 x′ = (sin2 t)x;
24.2 x′ = (sin(4t)− 1)x;
24.3 x′ = x+ y, y′ =cos t+ sin t
2 + sin t− cos ty;
24.4 x′ = −2x, y′ = sin(2t)x− 2y.
25. y²·K 50
26. y²·K 51 (b). 5¿: Ø^·K 51 (a) (Ø.
27. y²íØ 53 (d) Ú (e).
28. ¦ SL >¯K
y′′ = −λy,
y′(0) = 0, γy(x) + y′(1) = 0, γ > 0,
AÚA¼ê.
29. 3 SL >¯K (4.4.10) ¥?Úb q(t) ≤ 0, αβ > 0, γδ > 0. K (4.4.10) ¤kA
Ñ´K. J«: ^A¼ê¦±§ (4.4.9) ü>, ¿È©.
30. ¦e>¯K
y′′ = −λy,
y(−π) = y(π), y′(−π) = y′(π),
AÚA¼ê. 5¿µT>¯K¡±Ï>¯K. §Ø´ SL >¯K, é
AuÓA±kõ5Ã'Aþ.
31. ¦ Bessel § 2n ∈ Z Ï).
32. 3 Bessel §¥,
144
§4.5 SKo
(a) n ∈ N , y² J−n(x) = (−1)nJn (Ï J−n Jn ´5'), Bessel
¼ê÷v Poisson È©úª
Jn(x) =(2x)nn!
(2n)!π
∫ π
0
cos(x cos s) sin2n sds;
(b) é?¿ n > 0,
y2(x) = Jn(x)
∫dx
xJ2n(x)
,
´ Bessel § Jn(x) 5Ã').
33. Airy §
y′′ + xy = 0
´=IU©Æ[ George B. Airy ïÄ1Æ.
(a) ¦ Airy §?ê);
(b) y² Airy §3C u(t) = x−12 y(x), t = 2
3x32 e±=z¤ 1
3 Bessel §;
(c) Á^ 13 Bessel ¼êL« Airy §Ï).
34. y² Ricatti §
y′ = y2 + g(x)y + h(x),
3C u(x) = e−∫y(x)dx e±z¤àg5©§
u′′ − g(x)u′ + h(x)u = 0.
?Úy² Ricatti § y′ = y2 + x2 Ï)±L«¤
y(x, c) = xcJ− 3
4
(x2
2
)+ J 3
4
(x2
2
)J− 1
4
(x2
2
)− cJ 1
4
(x2
2
) ,½
y(x, c) = xJ− 3
4
(x2
2
)+ cJ 3
4
(x2
2
)cJ− 1
4
(x2
2
)− J 1
4
(x2
2
) ,Ù¥ c ∈ R ´?¿~ê.
35. y²
ex2 (t− 1
t ) =
∞∑n=−∞
Jn(x)tn.
145
1oÙ 5©§nØÚ)
¡þªàê1a Bessel ¼ê)¤¼ê. |^)¤¼êy²
2nJn(x) = xJn+1(x) + xJn−1(x),
J ′n(x) =1
2(Jn−1(x)− Jn+1(x)).
36. ^?ê)¦e§3 x = 0 ?ê)½Ï).
36.1. y′′ − xy = 0;
36.2. y′′ + x2y′ + 2xy = 0;
36.3. y′′ − 2xy′ + λy = 0, λ ∈ R ´ëê;
36.4. xy′′ + y′ + xy = 0;
36.5. x(1− x)y′′ − xy′ − 2y = 0;
36.6. x(1− x2)y′′ − 2y′ + xy = 0.
146
1ÊÙ ©§½5Ú½5nØ
§5.1 ©§)½5
3>f!Å!)Ô!Ôn!zÆ!7K$¬)¹¥éõ¢S¯K$Ä5ÆÑ
´d~©§½§|5£ã. ½©§|±káõ),§6u
ØÓЩ^. lØÓЩ^Ñu) ªªu,A½), ¡
). )353¢S¯K¥', §´©§½5nØïÄSN.
éu n ©§|
dx
dt= f(t,x), (5.1.1)
P x0(t) ´Ù÷vЩ^ x(t0) = x0 ).
• ¡) x0(t) ´½, XJé ∀ ε > 0, ∃ δ > 0 ¦ ‖x − x0‖ < δ , § (5.1.1) L
(t0,x)) x(t)Ñ÷v ‖x(t)−x0(t)‖ < ε, t ≥ t0. ù«½5q¡ Lyapunov ½.
• ¡) x0(t) ´ìC½, XJ§´½, éþã δ > 0, XJ ‖x − x0‖ < δ k
limt→∞
‖x(t)− x0(t)‖ = 0.
• ¡) x0(t) ´Ø½, XJ§Ø´½, = ∃ ε0 > 0 ¦é ∀ δ > 0 Ñ3 x∗δ ÷v
‖x∗δ − x0‖ < δ, 9é?¿ T ∗δ > 0 Ñ3m t∗δ > T ∗δ ¦ ‖x∗δ(tδ)− x0(t∗δ)‖ > ε0.
N5:
• $^ Lyapunov½5ïÄA½;½5,Ï~´²LCòÙ=z#§
²ï:5ïÄ.
• ìC½5½Â¥½bØUK. ~Xü ± S1 = z = e2π√−1θ, θ ∈ [0, 1)
þ©§
θ = sin2(πθ),
k²ï: θ = 0. l S1 þ?: (θ = 0Ø)Ñu; t→∞Ñ÷X
±_ªu²ï: 0, l ÷vìC½5½Â¥Ü©^, ²ï:
0 ´Ø½, ϧØ÷v½5½Â.
147
§5.1 ©§)½5
§5.1.1 àg5©§|")½5
½n58. A ∈M. éu~©§
y = Ay. (5.1.2)
(a) XJ A A¢ÜÑu", K§ (5.1.2) ")ìC½.
(b) XJ A A¢ÜÑu½u", ¢Ü"AêêuAÛ
ê, K§ (5.1.2) ") Lyapunov ½.
(c) XJ A k¢Üu"A, ½k¢Ü"AÙêêuAÛê,
K§ (5.1.2) ")ؽ.
y: (a) díØ 45 , 3 ρ > 0, a > 0 ¦é v ∈ Rn,
‖exAv‖2 ≤ a‖v‖e−ρx, x ∈ [0,∞).
§ (5.1.2) ?) y(x) ÑL«¤
y(x) = exAv,
/ª. ddy§ (5.1.2) ")ìC½.
(b) A pØÓA λ1, . . . , λk, λk+1, . . . , λs, §êê ni, i = 1, . . . , s;
Reλi < 0, i = 1, . . . , k, Reλi = 0, i = k + 1, . . . , s. Ï λi, i = k + 1, . . . , s êê
uÙAÛê, ¤±ÙéA5Ã'Aþk ni .
d½n 44 9ÙN5 1 , § (5.1.2)kÄ)Ý
Φ(x) =(eλ1xP
(1)1 (x), . . . , eλ1xP(1)
n1(x), . . . , eλsxP
(s)1 (x), . . . , eλsxP(s)
ns (x)),
Ù¥ P(j)i (x), j = 1, . . . , k, i = 1, . . . , nj ´gêØL nj − 1 õª, P
(j)i (x), j = k +
1, . . . , s, i = 1, . . . , nj ´~êþ (éAu λj Aþ). P
Ψ1(x) =(eλ1xP
(1)1 (x), . . . , eλ1xP(1)
n1(x), . . . , eλkxP
(k)1 (x), . . . , eλkxP(k)
nk(x)),
Ψ2(x) =(eλk+1xP
(k+1)1 , . . . , eλk+1xP(k+1)
nk+1, . . . , eλsxP
(s)1 , . . . , eλsxP(s)
ns
),
díØ 45 9Ùy², 3 ρ > 0, a > 0 ¦é ∀v1 ∈ Rn1+...+nk ,
‖Ψ1(x)v1‖2 ≤ a‖v1‖2e−ρx, x ∈ [0,∞).
148
1ÊÙ ©§½5Ú½5nØ
qÏ |eλix| = 1 (i = k + 1, . . . , s), P(i)j (i = k + 1, . . . , s, j = 1, . . . , ni) ´(½~êþ,
¤±3 b > 0 ¦é ∀v2 ∈ Rnk+1+...+ns ,
‖Ψ2(x)v2‖2 ≤ b‖v1‖2, x ∈ [0,∞).
§ (5.1.2)Ï)±¤
y(x) = Φ(x)v = Ψ1(x)v1 + Ψ2(x)v2,
Ù¥ v =
v1
v2
´?¿~êþ. ¤±
‖y(x)‖2 = ‖Φ(x)v‖2 ≤(a‖v1‖2e−ρx + b‖v1‖2
)≤ (a+ b)‖v‖2, x ∈ [0,∞).
ddy§ (5.1.2) ")´ Lyaponov ½.
(c) XJ A k¢Üu"A, P λ0 = α0 +√−1β0. d½n 44 , § (5.1.2)k/
X
y(x) = P(x)eλ0x,
), Ù¥ P(x) ´gêØL n− 1 õª. Ï α0 > 0, ¤±
limx→∞
‖y(x)‖2 = limx→∞
‖P(x)eλ0x‖2 = limx→∞
‖P(x)‖2eα0x =∞.
ùÒy²§ (5.1.2) ")ؽ
XJ A k¢Ü"A, P λ0 =√−1β0, ÙêêuAÛê. d½n
44 , § (5.1.2) k/X
y(x) = P(x)eλ0x,
), Ù¥ P(x) ´gêØ"õª. ¤±k
limx→∞
‖y(x)‖2 = limx→∞
‖P(x)‖2 =∞.
Ïd§ (5.1.2) ")ؽ. y..
íØ59. éu±Ï©§
y = A(x)y, (5.1.3)
Ù¥ A(x) ´ R þëY T ±Ï n ݼê.
149
§5.1 ©§)½5
(a) XJ§ (5.1.3) A¦êÑu 1 (½ Floquet ê¢ÜÑu"), KÙ"
)ìC½.
(b) XJ§ (5.1.3) A¦êÑu½u 1, 1 A¦êêê
uAÛê, KÙ")´ Lyapunov ½.
(c) XJ§ (5.1.3) kA¦êu 1, KÙ")ؽ.
y: Ùy²±d½n 48 (Floquet½n)Ú½n 58. [y²öSÖö¤.
y..
~K: q(t) ´ R þ±Ï T ±Ï¼ê. ½ Hill §
x′′(t) + q(t)x = 0,
±Ï)35Ú")½5.
): - x′ = y. Hill §=z
x′
y′
=
0 1
−q(t) 0
x
y
.
ù´±Ï T ±ÏXê5©§. b λ1, λ2 ´§üA¦ê. Kd·K 50
λ1λ2 = 1.
-
ρ =λ1 + λ2
2.
K
λ1 = ρ+√ρ2 − 1, λ2 = ρ−
√ρ2 − 1 = λ−1
1 .
e¡© ρ ØÓ?Ø Hill §)5.
1. ρ > 1. Kk
λ1 > 1 > λ2 > 0.
díØ 59 (c) , Hill §")ؽ. ?Ú/, Hill §kÏ)
x(t) = c1eµtp1(t) + c2e
−µtp2(t),
150
1ÊÙ ©§½5Ú½5nØ
Ù¥ µ = (lnλ1)/T > 0, p1(t), p2(t) ´ëY T ±Ï¼ê. ¯¢þ, Φ(t) ´ Hill §é
A§|Ä)Ý. -
C = Φ−1(0)Φ(T ), TR = ln C.
Kd Floquet ½n
Φ(t) = Q(t)etB,
Ù¥ Q(t) ´ T ±Ï¼ê. Ï3ÛÉÝ P ¦
B = Pdiag(eµ, e−µ)P−1,
¤±
Φ(t) = Q(t)Pdiag(eµt, e−µt)P−1.
|^Tª=Ï)Lª.
2. ρ = 1. Kk λ1 = λ2 = 1. d½n 49 (c), Hill §k² T ±Ï).
XJ C Jordan IO.´ü Ý, Kd 1 y²
Φ(t) = Q(t).
Ï z²)Ñ´ T ±Ï), l ")´½"
XJ C Jordan IO.Ø´ü Ý, K3ÛÉÝ P ¦
PCP−1 = E + N = eN,
Ù¥ N ´"Ý. -
B = P
(1
TN
)P−1.
Kk Φ(t) = Q(t)etB. qÏݼê
etB = PetT NP−1 = P
(E +
t
TN
)P−1,
t→∞ Ã., ¤±")´Ø½.
3. |ρ| < 1. Kk λ2 = λ1, λ2 éê´XJê. Ïdaqu 1 ¥y² Hill §Ï
)
x(t) = c1eµ√−1tp1(t) + c2e
−µ√−1tp2(t),
151
§5.1 ©§)½5
Ù¥ µ > 0, p1(t), p2(t) ´ëY T ±Ï¼ê. dTÏ)LªN´y² Hill §"
)´½. 5¿, d |λ1,2| = 1, íØ 59 ý")½5.
4. ρ = −1. Kk λ1 = λ2 = −1. d½n 49 (b), Hill §k² 2T ±Ï). d
Hill §kÄ)Ý Q(t)etB, Ù¥ Q(t) ´ 2T ±Ïݼê, B = (log C2)/(2T ). aqu
2y², C2 JordanIO.´(Ø´)ü Ý, Hill§")´½(ؽ
).
5. ρ < −1. Kk
λ2 < −1 < λ1 < 0.
díØ 59 (c) , Hill §")ؽ. d Hill §Ï)
x(t) = c1e−µ√−1tp1(t) + c2e
µ√−1tp2(t),
Ù¥ µ = (lnλ22)/(2T ) > 0, p1(t), p2(t) ´ëY 2T ±Ï¼ê.
N5: 'u Hill §\(J, k,Ööë [10, 42].
§5.1.2 d5Cq(½5§½5
5©§)½5¯K~(J. !ÄXÛÏL5©§)½
55(½5©§)½5.
½n60. éu5©§|
y = Ay + f(x,y), (5.1.4)
bA(x)3 [0,∞)þëY, f(x,y) = O(‖y‖2), ‖y‖ 1 3,« G = [0,∞)×y; ‖y‖ <
K þëY'u y ÷vÛÜ Lipschitz ^.
(a) XJ A A¢ÜÑu", K5©§ (5.1.4) ")ìC½.
(b) XJ A k¢ÜA, K5©§ (5.1.4) ")ؽ.
ÖòØÑT½ny². k,Öö±ë [53].
N5:
XJ Ak¢Ü"A,K5©§ (5.1.4)")½56u5
f(x,y) . ~X
152
1ÊÙ ©§½5Ú½5nØ
1. ©§|
x = −y + αx(x2 + y2)(x2 + y2 − 1),
y = x+ αy(x2 + y2)(x2 + y2 − 1),(5.1.5)
α = 0 , A5©§|XêÜ©A´éXJê, I:´
A5©§|¥%, Ï ´½, ìC½. α 6= 0 , ÏL4I
C x = r cos θ, y = r sin θ, §| (5.1.5) =z
r = αr3(r2 − 1), θ = 1.
dd§N´µ α > 0,lü ±SÜÑu;Ñ_^%C
I:, Ï §| (5.1.5) ")´ìC½. α < 0 , lü ±SÜ (I
:Ø) Ñu;Ñ_^%Cü ±, Ï §| (5.1.5) ")´
ؽ.
2. ©§|
x = −y − αy(x2 + y2)(x2 + y2 − 1),
y = x+ αx(x2 + y2)(x2 + y2 − 1),(5.1.6)
34IC x = r cos θ, y = r sin θ e=z
r = 0, θ = 1 + αr2(r2 − 1).
Ïdé α ?¿, §| (5.1.6) I:Ñ´¥%, ")´½Ø´ìC
½.
aq/±ÞÑ5©§XêÝäk"A, \þp
5©§")äk«U½5. ÖögCÑù~f. þã~f`²
5©§")5Ü©äk"¢ÜA, ")½5½¯K´
©(J.
§5.1.3 ½½5 Lyapunov 1
Äg£©§
dx
dt= f(x), x ∈ Rn. (5.1.7)
bXþ¼ê V (x), ‖x‖ ≤M ëY.
153
§5.1 ©§)½5
• ¼ê V (x) 'u§ (5.1.7) ê½Â
dV
dt
∣∣∣∣(5.1.7)
=∂V
∂x1x1 + . . .+
∂V
∂xnxn
∣∣∣∣(5.1.7)
=∂V
∂x1f1(x) + . . .+
∂V
∂xnfn(x).
• ¼ê V (x) ¡½ (½K) , XJ
V (0) = 0, V (x) > 0 (V (x) < 0), x 6= 0.
• ¼ê V (x) ¡~ (~K) , XJ
V (0) = 0, V (x) ≥ 0 (V (x) ≤ 0), x 6= 0.
• þã½Â¥¼ê V (x) Ú¡ Lyapunov ¼ê.
~K: ¼ê V (x1, x2) = x41 + x2
2 3 R2 ¥´½, 3 R3 ¥´~½. §'u
§
x1 = −x2, x2 = x1, (5.1.8)
ê
dV
dt
∣∣∣∣(5.1.8)
= 4x31(−x2) + 2x2x1 = 2x1x2(1− 2x2
1).
½n61. (Lyapunov ½5O) x = 0 ´§ (5.1.7) ). ¼ê V (x) 3 ‖x‖ ≤ M
þëY.
(a) XJ V (x) ´½, dVdt
∣∣(5.1.7)
´~K, K§ (5.1.7) ")´½.
(b) XJ V (x) ´½, dVdt
∣∣(5.1.7)
´½K, K§ (5.1.7) ")´ìC½.
(c) XJ V (x) ´½, dVdt
∣∣(5.1.7)
´½, K§ (5.1.7) ")´Ø½.
y: (a) é ∀ ε > 0 (Ø ε < M), -
m = minε≤‖x‖≤M
V (x).
Ï V ëY, V (x) 6= 0, ‖x‖ > 0, ¤± 0 < m <∞. qÏ V (0) = 0, ¤±3 δ > 0 ¦
V (x) < m, ‖x‖ < δ.
154
1ÊÙ ©§½5Ú½5nØ
eyé?¿ ‖x‖ < δ, § (5.1.7) 3, t0 l x Ñu) x(t) t > t0 Ñk
‖x(t)‖ < ε.
eØ,, 3 t∗ > t0 ¦ ‖x(t∗)‖ = ε, ‖x(t)‖ < ε, t ∈ [t0, t∗). d½n (a) b
V (x(t∗))− V (x(t0)) =
∫ t∗
t0
dV (x(t))
dt≤ 0.
‖x(t0)‖ < δ, ¤±
V (x(t∗)) ≤ V (x(t0)) < m.
ù V (x(t∗)) ½Âgñ. ùgñ`² ‖x‖ < δ, Òk ‖x(t)‖ < ε, t > t0. Ï ")
´½.
(b) d (a), § (5.1.7) ")´½. éu (a) y²¥½ δ. é ∀x ÷v ‖x‖ < δ,
du
dV (x(t))
dt≤ 0, V (x(t)) ≥ 0,
¤±4 limt→∞
V (x(t)) 3, PÙ l.
ey l = 0. eØ,, l > 0. Ï
V (x(t)) ≥ l, t ∈ [t0,∞),
V (0) = 0, ¤± ∃ ρ > 0 ¦
‖x(t)‖ ≥ ρ, t ∈ [t0,∞)
Ï
dV (x(t))
dt< 0, t ∈ [t0,∞).
-
r := maxρ≤‖x‖≤M
dV (x)
dt
∣∣∣∣(5.1.7)
.
Kk −∞ < r < 0. ¤±
V (x(t))− V (x(t0)) =
∫ t
t0
dV (x(s))
dsds ≤ r(t− t0), t ∈ [t0,∞).
t→∞ , >k, m>ªuKá. ùgñ`² l = 0.
y limt→∞
‖x(t)‖ = 0. eØ,, Ï x(t); t ∈ [t0,∞) k., ¤±3üN4O:
tn ÷v limn→∞
tn =∞ ¦
limn→∞
x(tn) = x∗ 6= 0.
155
§5.1 ©§)½5
d V (x) Ú) x(t) ëY5
0 = limn→∞
V (x(tn)) = V (x∗) 6= 0.
ùgñ`² limt→∞
‖x(t)‖ = 0.
(c) y. XJ")´½, Ké ∀ ε > 0, ∃ δ > 0 ¦ 0 < ‖x‖ < δ , ÷vЩ^
x(t0) = x )k
‖x(t)‖ < ε, t ∈ [t0,∞).
Ï
dV (x)
dt
∣∣∣∣(5.1.7)
> 0, x 6= 0,
¤±
V (x(t)) ≥ V (x(t0)) = V (x) > 0.
l ∃σ > 0 ¦
‖x(t)‖ ≥ σ, t ∈ [t0,∞).
-
% = minσ≤‖x‖≤ε
dV (x)
dt
∣∣∣∣(5.1.7)
.
K 0 < % <∞. l k
V (x(t))− V (x(t0)) =
∫ t
t0
dV (x(s))
dsds ≥ %(t− t0).
Ï σ ≤ ‖x(t)‖ < ε V ëY, ¤± t→∞ , >k, m>ªÃ¡. ùgñ`
²")´Ø½. y..
N5:
1. ½n 61 ¥½")½5¡ Lyanunov 1. §´ÛdêÆ[ Alek-
sandr Lyapunov (1857–1918)u 1892c3ÙƬة¥JÑ ( [41]),T3Äå
XÚ½5ïÄ¡åX^. ؽn 61 ¥O, kÙ§
í2. ÖØ3dÛ.
2. éu1, Lyapunov 1^?ê½½5, du^å5ØB, y3é
Jå. Lyapunov 1' Lyapunov ê3yÄåXÚ·bïÄ¡
åX4Ù^.
156
1ÊÙ ©§½5Ú½5nØ
~K:
1. §
x = x3 − 2y3, y = xy2 + x2y +1
2y3,
")ؽ, ±ÏL Lyapunov ¼ê V = x2 + 2y2 y.
2. §
x = y − x3, y = −2(x3 + y5),
")ìC½, ±ÏL Lyapunov ¼ê V = x4 + y2 y.
3. §
x = y + 2y3, y = −x− 2x3,
")½, ±ÏL Lyapunov ¼ê V = x2 + x4 + y2 + y4 y.
4. §
x = 2x2y + y3, y = −xy2 + 2x5, (5.1.9)
")ؽ. ¯¢þ, ¼ê V = xy, K
dV
dt
∣∣∣∣(5.1.9)
= x2y2 + y4 + 2x6, (5.1.10)
3")S½. du3 y ¶þ x > 0, 3 x ¶þ y > 0, ¤±l1
Ñu;XmO\ò©ª±31. ql (5.1.10) , ÷Xl1
¥?:Ñu;XmO\ V = xy îO\,ù`²l1
Ñu;XmO\ÑòÅìl"). l ")´Ø½.
N5: äNA^ Lyapunov 1'´EÜ· Lyapunov ¼ê, vk
E Lyapunov ¼ê. éuäN¯K, 3E Lyapunov ¼ê¦þ¦Ù÷X
§ê¥Ø¹CþÛg.
§5.2 ²¡g£©XÚ: 4Ú©|
üå, IJ¡g£©XÚ
x = P (x, y), y = Q(x, y), (5.2.1)
157
§5.2 ²¡g£©XÚ: 4Ú©|
Ù¥ P, Q 3,²¡« Ω ¥ëY. P, Q ´5¼ê, ©XÚ (5.2.1) )Ú;
531oÙ¥®)û.
P,Q ´gêu½u 2 õª, ©XÚ (5.2.1) ÄåƯKvk)û. Ï
P,Q ´1w¼ê½)Û¼ê, ©XÚ (5.2.1) ÄåÆïÄ\(J. 5
`,é©XÚ (5.2.1)^ÐÈ©¦)´ØU. 19V"IêÆ[ Henry Poincare
Mᩧ½5nØ, ©§ïÄm÷Ï#. ϱ5©§±
Ï)ïÄ´²¡½5nØØ%K, ë [3, 12, 18, 19, 25, 30, 34, 39, 40, 45, 47,
48, 57, 58, 60–62].
b P, Q 3« Ω ⊂ R2 ¥ëY, ©XÚ (5.2.1) 3 u Ω ¥k±Ï; Γ.
• ¡ Γ TXÚ4, XJ3 Γ ?¿ýÚSýÑk4;.
• ¡ Γ TXÚ½4, XJl Γ ,S?:Ñu;Ñ^%
C Γ.
• ¡ Γ TXÚؽ4, XJl Γ ,S?:Ñu;ÑK^
%C Γ.
• ¡ Γ TXÚ½4, XJl Γ ,ýÑu;Ñ^%C
Γ, l,ýÑu;ÑK^%C Γ.
• ¡ Γ TXÚ±Ï, XJ Γ ,¥¿÷±Ï;.
~K: ²¡4Ie©§
r = r(r − 1)(r − 2)2(r − 3), θ = 1,
±Ï; r = 1 ´½4, r = 2 ´½4, r = 3 ´Ø½4.
N5:
• ©XÚ (5.2.1) )Û, 31w©XÚ~fµ§,±Ï;?¿
SÑQk±Ï;qk±Ï;. k,Ööë^ [57, p.44 ~ 2].
• ©XÚ (5.2.1) )Û, |^)Û¼ê":5±y²§?±Ï;½ö
k¿÷±Ï;,½ökÙ¥vkÙ§±Ï;. ö¡á
±Ï;. Ïdéu)Û©XÚ, 4q½Âá±Ï;.
158
1ÊÙ ©§½5Ú½5nØ
N5: Hilbert 1 16 ¯K
• 1900c3niISêÆ[¬þ,IêÆ[ David HilbertJÑ 23êƯK.Ù
¥1 16¯KÜ©¯µ©XÚ (5.2.1) ´¢ ngõª©XÚ (= P,Q´
'u x, y ¢Xêõª, maxdegP,degQ = n) , §4êÚ©
Ù? ë [27, 50].
• 100 õcL, ¦+(J, lT¯K)ûé. ~X
– 3 1990 ccIêÆ[ Ecalle [19] ÚÛdêÆ[ Ilyashenko [30] ©Oy²µ
?¿½²¡ n gõª©XÚ4ê´k.
– é¤k²¡ ngõª©XÚ4ê´Äkþ.¯Kvk)û
(=¦´²¡gõª©XÚvk)û).
þã¯KïÄ(Jo(3;ÍÚnãw¥,k,Öö±ë [12, 18, 34,
39, 47, 48, 57, 58, 61].
§5.2.1 435Ú½5½
ÄkÑAü½4Ø3Ú3(Ø.
½n62. (Bendixson ½n) e©XÚ (5.2.1) uÑþ
div(P,Q) =∂P
∂x+∂Q
∂y,
3üëÏ« Ω ¥±~Ò, Ø3 Ω ?Ûf«þðu", K©XÚ (5.2.1) 3 Ω
¥vk4;, l vk4.
y: y. e Γ ´ u Ω ¥^4;. D ´ Γ ¤«SÜ. Kd Green úª
9½nb
0 6=∫∫
D
(∂P
∂x+∂Q
∂y
)dxdy =
∮Γ
Pdy −Qdx = 0.
ùgñL²©XÚ (5.2.1) 3 Ω ¥vk4;. y..
N5:
159
§5.2 ²¡g£©XÚ: 4Ú©|
• lþã½ny²µ©XÚ (5.2.1) 3 Ω ¥k4; Γ 7^´∫∫D
(∂P
∂x+∂Q
∂y
)dxdy = 0.
l½n 62 y²éN´e(J.
íØ63. (Dulac ½n) e3üëÏ« Ω ¥3ëY¼ê B(x, y) 6= 0 ¦
∂(BP )
∂x+∂(BQ)
∂y,
3 Ω ¥±~Ò, 3 Ω ?Ûf«þØðu", K©XÚ (5.2.1) 3 Ω ¥vk4
;, l vk4.
íØ 63 ¥¼ê B ¡ Dulac ¼ê. ¦+íØ 63 ½n 62 ké«O, 3
4Ø3ïÄþ%å~^. Ïéõ¹e, XÚ (5.2.1) ¿Ø÷
v½n 62 ^, %3¼ê B ¦íØ 63 ^¤á. äN¯K¥é B %´(J.
e¡(JÑ43OOK.
½n64. (Poincare–Bendixson½n) Ω ´. e©XÚ (5.2.1) 3 Ω ¥vkÛ
:, §; Ω S>.Ñ (½ÑK) ?\ Ω SÜ, K Ω ¥k
^¹S¸.3ÙSÜ«½ (ؽ) 4Ú^S½ (ؽ) 4.
?Ú?Ø435Ú½5, IïÄ Poincare N. Γ ´©XÚ
(5.2.1) ^±Ï;.
• é ∀p = (x, y) ∈ Γ,L p1wã S ¡©XÚ (5.2.1)è,XJé ∀q ∈ S,
þ (P (q), Q(q)) S î (=Ø S 3T:þ²1).
• é ∀q ∈ S, § (5.2.1) L q ;P φt(q). XJ q p ålv, Kd)'
uЩ^ëY65, φt(q) ;ò£ S þ. P T (q) L q ;
1g£ S m, ¡£m.
• Σ ⊂ S ¦é ∀q ∈ Σ k φT (q)(q) ∈ S. ¡ Σ Γ Poincare è. ùÒ½Â
N
P : Σ −→ S
q −→ φT (q)(q).
¡ Σ þ Poincare N½£N.
160
1ÊÙ ©§½5Ú½5nØ
N5: éu©XÚ (5.2.1),
• Poincare N P(q) ´N, §ØÄ:éAX©XÚ (5.2.1) ±Ï;.
• p ∈ Γ ´ Poincare N P ØÄ:, Γ ½5dN P 3ØÄ: p ∈ Γ A λ
(½. λ < 0 , Γ ´½; λ > 0 , Γ ´Ø½.
• Poincare N´8cïıÏ;35Ú½5Ìóä.
e¡(J`² Poincare N©XÚkÓ1w5.
·K65. ©XÚ (5.2.1) ±Ï;£mÚ Poincare N©XÚkÓ1
w5.
y: Γ ´©XÚ (5.2.1) ^±Ï;. S v1w Γ ^è. d)'u
ÐÚëêëY65, ©XÚ (5.2.1) ) φt(q) TXÚkÓ1w5. Ïdé
q ∈ S, Iy² T (q) ©XÚ (5.2.1) kÓ1w5.
S ëêL« (w, g(w)), K g S kÓ1w5. éu S ¥:
q = (w, z), -
F (q) = z − g(w).
K
q ∈ S ⇐⇒ F (q) = 0.
-
G(t,q) = F (φt(q)).
Ké q ∈ S,
G(0,q) = F (φ0(q)) = F (q) = 0,
∂G
∂t
∣∣∣∣(0,q)
=
(∂F
∂w(q),
∂F
∂z(q)
)dφ(t)
dt
∣∣∣∣t=0
= (−g′(w), 1) · (P (q), Q(q)) 6= 0,
Ù¥ ·:L«þSÈ.ت¤á,Ï (−g′(w), 1)´ S 3 (w, z) = (w, g(w))
FÝ, þ| (P,Q) S î.
dÛ¼ê3½nµ3 G kÓ1w5¼ê t = t(q) ¦
G(t(q),q) = F (φt(q)(q)) ≡ 0, q ∈ S.
161
§5.2 ²¡g£©XÚ: 4Ú©|
l T (q) ©XÚ (5.2.1) kÓ1w5. y..
e¡(JÑ4½5½.
½n66. Γ := φ(t) = (x(t), y(t)), t ∈ R ´©XÚ (5.2.1) ±Ï;, ±Ï T . X
JXÚuÑþ÷X±Ï;È©∫ T
0
(∂P (φ(t))
∂x+∂Q(φ(t))
∂y
)dt < 0 (> 0)
K Γ ´½ (ؽ) 4.
y: ½ny²Ìg^ [57]. ½ Γ þ?: p0, b Γ ´K½, =Xm
O\l p0 Ñu;^$Ä. é ∀p ∈ Γ, Pl p0 p l s, ^
(±eÄ^). Γ l l, Γ ±l s ëêëê§
x = φ(s), y = ψ(s), s ∈ [0, l].
K
φ′(s) =x
s=
P0√P 2
0 +Q20
, ψ′(s) =y
s=
Q0√P 2
0 +Q20
,
Ù¥ x L« x 'um t ê,
P0 = P (φ(s), ψ(s)), Q0 = Q(φ(s), ψ(s)).
éu Γ ,¿©S?: q, ½Â§ÛÜI (s, n) Xe: q u Γ þ
,: p þ, s ´l p0 p Ý, n ´l p q ÷XÝ. |^ÛÜI
(s, n), q IL«
x = φ(s)− nψ′(s), y = ψ(s) + nφ′(s), (5.2.2)
Ù¥ n c¡ÎÒ´d φ′(s) Ú ψ′(s) ÎÒ(½. ùp^ Γ L p Y²
YuÚu©O´
Q0√P 2
0 +Q20
ÚP0√
P 20 +Q2
0
.
e¡F"ïÄl Γ¿©S: qÑu;$ı nCz. 5¿3
T;þ n ´ s ¼ê, ;§ (5.2.2). ò (5.2.2) \©XÚ (5.2.1)
Q[s, n]
P [s, n]=dy
dx=ψ′(s) + n′(s)φ′(s) + nφ′′(s)
φ′(s)− n′(s)ψ′(s)− nψ′′(s),
162
1ÊÙ ©§½5Ú½5nØ
Ù¥ [s, n] = (φ(s)− nψ′(s), ψ(s) + nφ′(s)). )d§
n′(s) =Q[s, n]φ′(s)− P [s, n]ψ′(s)− n(s)(P [s, n]φ′′(s) +Q[s, n]ψ′′(s))
P [s, n]φ′(s) +Q[s, n]ψ′(s)=: F (s, n).
w, n ¿©, P [s, n]φ′(s) +Q[s, n]ψ′(s) 6= 0, s ∈ [0, l], F (s, 0) = 0. é F (s, n) 'u
n Taylor Ðm
n′(s) = H(s)n+ o(n), (5.2.3)
Ù¥
H(s) =∂F
∂n
∣∣∣∣n=0
=P 2
0Q0y − P0Q0(P0y +Q0x) +Q20P0x
(P 20 +Q2
0)3/2,
´± l ±Ï±Ï¼ê, Ù¥ f0x, f ∈ P,Q, L« f0 'u x ê.
é§
n′(s) = H(s)n, (5.2.4)
È©
n(l) = n(0)e∫ l0H(ν)dν .
¤±
D :=
∫ l
0
H(ν)dν < 0 (> 0),
§ (5.2.4) ")ìC½ (ؽ). l D < 0 (> 0) , § (5.2.3) ")ìC
½ (ؽ), Ï Γ ìC½ (ؽ).
Ï÷X±Ï; Γ k ds =√P 2
0 +Q20dt, ¤±ÏLO∫ l
0
H(ν)dν =
∫ T
0
P 20Q0y − P0Q0(P0y +Q0x) +Q2
0P0x
P 20 +Q2
0
dt
=
∫ T
0
[P0x +Q0y −
P 20P0x + P0Q0(P0y +Q0x) +Q2
0P0y
P 20 +Q2
0
]dt
=
∫ T
0
(P0x +Q0y) dt− 1
2
∮Γ
d(P 2
0 +Q20
)P 2
0 +Q20
dt =
∫ T
0
(P0x +Q0y) dt.
ùÒy²½n(Ø. y..
N5: ±þ04nØ¡AÄ£, éT¯Kk,ÖöÖ [40,
57, 60, 61].
163
§5.2 ²¡g£©XÚ: 4Ú©|
§5.2.2 ©|¯KAü~f
©|nØäk4Ù´LSN.oÑ/`, §ÌïÄ©XÚëêCz©X
ÚN½ÛÜ;(Cz (ÿÀ¿Âe).
!ÏLü~f50ÄåXÚ¥A~©|y.éu²¡©XÚ (5.2.1),
P J = ∂(P,Q)∂(x,y) Ù Jacobi Ý.
• p = (x0, y0) ¡ (5.2.1) Û:, XJ P (p) = Q(q) = 0.
• Û: p ¡©XÚ (5.2.1) VÛ:, XJ J 3 p üA¢ÜÑØ".
• Û: p ¡©XÚ (5.2.1) ÐQ: (½VQ:), XJ J 3 p küÉÒA
.
• Û: p ¡©XÚ (5.2.1) Ð(: (½V(:), XJ J 3 p küÓÒA
.
• Û: p ¡©XÚ (5.2.1) o: (¡:), XJ J 3 p kéÝEA,
¢ÜØ".
• Û: p ¡©XÚ (5.2.1) ¥%, XJ p ,¿÷TXÚ±Ï;.
• Û: p ¡©XÚ (5.2.1) [:, XJ p Ø´¥%, J 3 p kéÝXJ
A.
• ©XÚ (5.2.1) ^;¡Ó;, XJ§ÚKѪuÓÛ:.
• ©XÚ (5.2.1) ^;¡É;, XJ§ÚK©OªuØÓÛ:.
~K:
1. IJ¡©XÚ
x = −y − x(x2 + y2 − 2√x2 + y2 + α),
y = x− y(x2 + y2 − 2√x2 + y2 + α),
α ∈ R. (5.2.5)
N´yµ α > 0 , :´½:; α < 0 , :´Ø½
:. α = 0 , XÚ (5.2.5) 3Û: (0, 0) Jacobi ÝkéXJA.
164
1ÊÙ ©§½5Ú½5nØ
34IC x = r cos θ, y = r sin θ e, ©XÚ (5.2.5) =z¤
r = −r(r2 − 2r + α), θ = 1.
lT§N´©XÚ (5.2.5) e(Ø:
– α > 1. :´½:. Ï r < 0, r 6= 0, ¤±l²¡þ?:Ñu;
Ñ^%Cu:.
– α = 1. XÚk½4 r = 1, §Sܴؽ, Ü´½.
d¡)½©|.
– 0 < α < 1. ½©|Ñü4µr = 1−√
1− α Ú r = 1 +√
1− α, Ù¥
SÜ4´Ø½, Ü4´½. X α ~, SÜ4
Øä , Ü4Øä*.
– α = 0. SÜؽ4 :¦Ù¤Ø½[:. d¡
) Hopf ©|.
– α < 0. :´Ø½o:, XÚk½4.
2. ²¡©XÚ
x = −y − α(x2 + y2)(x2 + y2 − 1)(x2 + y2 − 4),
y = x− α(x2 + y2)(x2 + y2 − 1)(x2 + y2 − 4),α ∈ R. (5.2.6)
34IC x = r cos θ, y = r sin θ e=z
r = −αr2(r2 − 1)(r2 − 4), θ = 1.
N´y
– α = 0 , :´Û¥%.
– α 6= 0 , XÚkü4: r = 1 Ú r = 2. ¡d¥%©|Ñ4,
ù«©|y¡ Poincare ©|.
3. ²¡©XÚ
x = 2y, y = 2x+ 3x2, (5.2.7)
165
§5.2 ²¡g£©XÚ: 4Ú©|
´ Hamilton ©XÚ (Ï È), Ù Hamilton ¼ê
H = y2 − x2 − x3.
Hamilton XÚ (5.2.7) ± (− 23 , 0) ¥%, y2 − x2 − x3 = 0 ¹Q: (0, 0) ÚªuT
Q:Ó;.
4. ²¡©XÚ
x = 2y, y = a2 sinx, a > 0, (5.2.8)
´ Hamilton ©XÚ, Ù Hamilton ¼ê
H = y2 + a2 cosx.
ª H = −a2 ¥¹kÉuQ: (−π + 2kπ, 0) Ú (π + 2kπ, 0), k ∈ Z, É;
. (2kπ, 0), k ∈ Z ´¥%.
N5:
• ~ 1¥XÚëêCz,XÚ¬â,aѽ.ù´4ïÄ¥Ì(J
. ¢Sþ, ÖöéN´EÑù~fµ3ëêCzL§¥XÚ¬â,aÑ?¿
ê4, ¿©¤õ4.
• 3~ 1¥,ëê αlKO\,:½5u)Cz,l )4.
ù)4¡dHopf ©|)4. d Hopf ©|)4´8cï
Ä4¯KÃã.
• éu¹ëê©XÚ,ëêCz,lXÚÓ;©|Ñ4¡Ó©|.
lXÚÉ;©|Ñ4¡É©|.
• ~ 2 ¥ Poincare ©|´8cïÄ4¯KÌÃã. ïÄlÈ©X
Ú¥%9Ù>.©|Ñ4ê8¯K¡f Hilbert1 16¯K,ë [12].
ÏLü~f0AÛ:©|.
~K:
1. Q(:(saddle-node)©| ¹ëê λ ©§
x′ = x2 + λ, λ ∈ R.
166
1ÊÙ ©§½5Ú½5nØ
– λ > 0 vkÛ:;
– λ = 0 kÛ:, §´½, ¡Q(:;
– λ < 0 þãÛ:©¤üÛ:: ½, ؽ.
ù«üÛ:Üy¡Q(:©|.
2. .(Transcritical)©| ¹ëê λ ©§
x′ = x2 − λx, λ ∈ R.
– λ > 0 küÛ:: Û: x = 0 ½, Û: x = λ ؽ;
– λ = 0 üÛ:3 x = 0 ܤ½Û:;
– λ < 0 þãÛ:©¤üÛ:: x = 0 ؽ, x = λ ½.
ù«üÛ:ܽ5y¡.©|.
3. ú(Pitchfork)©| ¹ëê λ ©§
x′ = x(x2 + λ), λ ∈ R.
– λ > 0 kؽÛ:: x = 0;
– λ = 0 knÛ:µx = 0, §E,´½;
– λ < 0 þãnÛ:©¤nÛ:: x = −√−λ ½, x = 0 ؽ,
x =√−λ ½.
ù«3ëêCz,L§¥±Û:, ,â,©¤nÛ:y¡ú
©|.
4. ´¢(Hysteresis)©| ¹ëê λ ©§
x′ = x3 − x− 2λ, λ ∈ R.
– λ > 3−3/2 kؽÛ:, P x = x1;
– λ = 3−3/2 küÛ:µØ½Û: x = x1 ÚQ(:, P x = x0, Kk
x0 < x1;
167
§5.3 SKÊ
– −3−3/2 < λ < 3−3/2 Q(: x0 ©¤ x = x2 Ú x = x3. dnÛ:
x3 < x2 < x1 ©Oؽ!½Úؽ;
– λ = −3−3/2 x = x2 x = x1 ܤQ(:;
– λ < −3−3/2 Q(:, kؽÛ: x = x3.
ù«3ëêCzL§¥Q(:â,aÑ!©!2Ù§Û:(ܤQ(:â,
y¡´¢©|.
5. Q(:©| ¹ëê λ ©§
x′ = x2 − λ, y′ = y, λ ∈ R.
– λ < 0 vkÛ:;
– λ = 0 kÛ: (0, 0), ¡Q(:;
– λ > 0 þãÛ:©¤üÛ:: (√λ, 0) ؽ(:, (−
√λ, 0) Q:.
ù«â,aÑÛ:, ,©¤Q:(:y¡Q(:©|.
F")©|¯KnØ£Ú?Ú(J,k,Ööë [11, 12, 18, 22, 29,
31, 32, 36, 39, 47, 48, 62].
§5.3 SKÊ
1. y²íØ 59.
2. ÏL¦e§A¦ê½")½5
2.1 x′ = 4 sin(3t)x;
2.2 x′ = −2x+ sin(2t)y, y′ = −2y.
2.3 x′ = x+ cos(2t)y, y′ = (1 + cos(2πt))y.
3. Mathieu §
x′′ + (a+ b cos t)x = 0
´ Hill §AϹ, Ù¥ a, b ´~ê. ÁÒ a, b ØÓ, ½ Mathieu §")
½5Ú±Ï)35.
168
1Ù ©§½5Ú½5nØ
4. ½e§|")½5
4.1 x = 2x− y + x2y + 3y3 − 5y5, y = x− y + x2 − y2 + 2xy.
4.2 x = −2x− y + xy − y3 + 3xy2, y = −y + xy2 + 2x3.
4.3 x = 2x− 3y + x2 + 3y2, y = 3x+ 2y + xy2 − y3 + x2y2.
4.4 x = −y − αx(x2 + y2 − 1), y = x− αy(x2 + y2 − 1).
5. n?>f+¥g-§
x+ ε(x2 − 1)x+ x = 0,
´Ö=ÔnÆ[ Balthasar van der Pol u 1926 cïá (<¡ van der Pol
§). T§gïá±53Ôn!)ÔÆ¥k2A^. Á?Ø van der Pol §
")½5. J«µ=z§|?Ø.
6. ÁÑ5XÚ~fµ§")ìC½, Ù5CqXÚ")ؽ.
7. ^ Lyapunov 1½e§|")½5.
7.1 x = −y + αx5, y = x+ αy5.
7.2 x = −x+ 2xy2, y = −4x2y − 3y5.
7.3 x = 2x2 − y2, y = xy. J«µ V (x, y) = x2 − y2.
7.4 x = x3 − 2y3, y = xy2 + 2x2y + y3.
7.5 x = y − 2x− 3x3y2, y = −5x− 7y.
7.6 x = y + 2y3, y = −x− 2x3.
8. b f(x, y) ∈ C1(R2). Á?ا|
x = y − xf(x, y), y = −x− yf(x, y),
")½5.
9. b f(x) ∈ C(R), xf(x) < 0, x 6= 0. Á?ا|
x = y, y = f(x),
")½5.
169
§5.3 SKÊ
10. ½§|
x = −2y, y = 2x+ 3x2,
¤kÛ:½5.
11. ½²¡©XÚ
x = −y + αx(x2 + y2 − 1)2(x2 + y2 − 4), y = x+ αy(x2 + y2 − 1)2(x2 + y2 − 4)
´Äk4. ek, ѧLª, ¿½§½5.
12. y²íØ 63.
13. ©Ûe¹ëꩧ©|, ¿xÑ©|ã.
13.1 x′ = x3(x− λ).
13.2 x′ = x2 − αx+ β.
13.3 x′ = x3 − λx+ 2.
13.4 x′ =x2
1 + x2− λ.
13.5 x′ = µx− y − βx(x2 + y2), y′ = x+ µy − βy(x2 + y2).
13.6 x′ = y, y′ = −x(1− x) + λ(1− x)y.
170
18Ù N¹
§6.1 Arzela-Ascoli Úny²
½n67. (Arzela–Ascoli Ún) XJ¼ê fn(x) 3k.48 I þk.ÝëY,
K fn(x) kf3 I þÂñ.
y: P I = [a, b]. Äky²(Ø 1µéu I þ?¿k: x1, . . . , xm, o3 fn(x)
f¦§3ù m :z:Ñ´Âñê.
¯¢þ,du fn(x)3 Iþk.,¤± fn(x1)3Âñf. P fnk1 (x1)∞k1=1
fn(x1)Âñf. q fnk1 (x)3 I þk.,¤± fnk1 (x2)3Âñf
, P fnk2 (x2)∞k2=1. aq/, fnk2 (x) 3 I þk., ¤± fnk2 (x3) 3Âñf
, P fnk3 (x3)∞k3=1. ù= fn(x) f¦§3ù m :Ñ´Âñ. l
(Ø 1 ¤á.
é ∀ ε > 0, du fn(x) 3 I þÝëY, ¤± ∃ δ > 0 ¦é ∀z1, z2 ∈ I,
|x1 − x2| < δ, Òk
|fn(z1)− fn(z2)| < ε, ∀n ∈ N. (6.1.1)
éuþã ε, δ, - m = [ b−aδ ] + 1. ò«m I = [a, b] ©¤ m °, ò§ m+ 1 à:
lmgP x0, x1, . . . , xm. Kz«mÝÑu δ.
d(Ø 1 , ¼ê fn(x) 3f fnk(x) 3 x0, x1, . . . , xm ÑÂñ. P fnk(x)
3ù m+ 1 :4©O f(x0), f(x1), . . . , f(xm).
Ùgy²(Ø 2: fnk(x) 3 I þÂñ. ¯¢þ, éuþã?¿À ε > 0, du
fnk(xi), i = 0, 1, . . . ,m, Âñu f(xi), ¤± ∃Ni ∈ N ¦ l > Ni ,
|fnl(xi)− f(xi)| <ε
8. (6.1.2)
- N = maxN0, N1, . . . , Nm. K l > N k
|fnl(xi)− f(xi)| <ε
8, ∀ i ∈ 0, 1, . . . ,m (6.1.3)
171
~©§
é ∀x ∈ I, K ∃ i ∈ 0, 1, . . . ,m ¦ x ∈ [xi, xi+1]. l é ∀ l1, l2 > N k
|fnl1 (x)− fnl2 (x)| ≤ |fnl1 (x)− fnl1 (xi)|+ |fnl1 (xi)− f(xi)| (6.1.4)
+|f(xi)− fnl2 (xi)|+ |fnl2 (xi)− fnl2 (x)| < ε
2.
= fnk(x) ´ Cauchy ê, Ï Âñ. PÙ4 f(x).
éþã?¿À½ ε > 0, 9þã N > 0. é ∀x ∈ I, 9 l, k > N , d (6.1.4)
|fnl(x)− fnk(x)| < ε
2.
éþª- k →∞
|fnl(x) − f(x)| ≤ ε
2< ε.
ùÒy² fnk(x) 3 I Âñ f(x). y..
5µdëY¼êÂñ5, fnk(x)3 I Âñ4¼ê f(x)3k.48
I þëY.
§6.2 Ýéê35y²
½n68. 'uÝéê, e(ؤá.
(a) XJ C ´ n ÛÉÝ (¢½E), K3 n Ý B ¦ C = eB. ¡
B ´ C Ýéê.
(b) XJ C ´ n ÛÉ¢Ý, K3 n ¢Ý B ¦ C2 = eB.
yµ(a) dÝ Jordan IO.nØ, 3 n ÛÉÝ P ¦ P−1CP = J ´ Jordan
IO.. XJ3 n Ý K ¦ J = eK, K B := PeKP−1 = C Ýéê. Ïde
¡Iy² Jordan IO. J kÝéê.
- J = diag(J1, . . . ,Jm), Ù¥ Ji ´ Jordan ¬. âÝê5, Iy²z
Jordan ¬kÝéê. ÏdPÒüå, Øb J ´ Jordan ¬. -
J = λE + N = λ(E + λ−1N),
Ù¥ N ´"Ý. |^ ln(1 + x) Taylor Ðm
ln J = (lnλ)E +
n−1∑k=1
(−1)k+1
k(λ−1N)k, (6.2.1)
172
18Ù N¹
Ù¥^"Ý5 Nk = 0, k ≥ n. Ïd K = ln J ´ J éê.
(b) du C ´¢, ¤±3Ûɢݦ C2 qu§¢ Jordan IO.. Ïd
aqu (a) y², Iy² C2 Jordan IO.k¢éê.
λ1, . . . , λn ´ CA,K λ21, . . . , λ
2n ´ C2 A, C2 JordanIO.¥
Jordan ¬éAu C Jordan IO.¥ Jordan ¬. Au C ØÓa.A, «
©en«¹.
1. Au C ¢A λ éA C2 Jordan ¬ J. d
J = λ2Ei + Ni,
Ù¥ Ei Ú Ni ´·êü ÝÚ¢"Ý. l (6.2.1) , ¢ Jordan ¬ J k¢
Ýéê.
2. Au C XJA λ =√−1b éA C2 Jordan ¬ J. d
J =
−b2
1 −b2
. . .. . .
1 −b2
.
du C ´¢, §EA¤éÑy. Ï C2 Au C XJA −√−1b
Jordan ¬´ J. ù3 C2 Jordan IO.¥k¬ M = diag(J,J). Ø J ´ l Ý
. ey M k¢Ýéê.
ÄkN´y² M qu
Λ + N,
Ù¥
Λ =
−b2E2
−b2E2
. . .
−b2E2
, N =
0
E2 0
. . .. . .
E2 0,
,
Ù¥ 0 ´"Ý, E2 ´ü Ý. ¤±
ln(Λ + N) = ln Λ + ln(E2l + Λ−1N) = ln Λ +
n−1∑k=1
(−1)k+1
k(Λ−1N)k,
173
~©§
Ù¥1ª^ Λ−1N ´"Ý. q
ln Λ =
ln(−b2E2)
ln(−b2E2)
. . .
ln(−b2E2)
,
ln(−b2E2) =
ln b2 (2k + 1)π
−(2k + 1)π ln b2
, k ∈ Z.
ùÒy² M, l J k¢éê.
3. Au C EJA λ = a+√−1b éA C2 Jordan ¬ J, ab 6= 0. d
J =
D2
E2 D2
. . .. . .
E2 D2
, D2 =
a b
−b a
.
aqu 2 ¥y², ò J ¤
J = Λ + N,
Ù¥
Λ =
D2
D2
. . .
D2
, N =
0
E2 0
. . .. . .
E2 0,
.
du
ln D2 =
1
2ln(a2 + b2) arccos
(a√
a2 + b2
)+ 2lπ
− arccos
(a√
a2 + b2
)− 2lπ
1
2ln(a2 + b2)
, l ∈ Z,
aqu 2 y², J k¢éê.
nÜþãn«¹y, C2 ¢ Jordan IO.k¢éê, Ï C2 k¢éê. y
..
174
ë©z
[1] Abraham R. and Marsden J.E., Foundations of Mechanics, The Benjamin/Cummings
Publishing Company, INC, Lodon, 1978.
[2] Arnold V.I., Ordianry Differential Equations, Springer-Verlag, Berlin, 2006.
[3] Arnold V.I., Geometric Method of the Theory of Ordianry Differential Equations,
Springer-Verlag, New York, 1983.
[4] Arnold V.I., Mathmatical Methods of Classical Mechanics, Springer-Verlag, New York,
1989.
[5] Audin M., da Silva A.C. and Lerman E., Symplectic Geometry of Integrable Hamiltonian
Systems, Birkhauser, Basel, 2003.
[6] Bogoyavlenskij O.I., Extended integrability and bi-Hamiltonian systems, Commun.
Math. Phys. 196 (1998), 19-51.
[7] Bolsinov A.V. and Fomenko A.T., Integrable Hamilton Systems: Geometry, Topology,
Classification, Chapman & Hall, Boca Raton, 2004.
[8] Bolsinov A.V. and Taimanov I. A., Integrable geodesic flows with positive topological
entropy, Invent. Math. 140 (2000), 639-650.
[9] Chavarriga J., Giacomini H., Gine J. and Llibre J., On the integrability of two-
dimensional flows, J. Differential Equations 157 (1999), 163-182.
[10] Chicone C., Ordianry Differential Equations with Applications, Springer-Verlag, New
York, 2006.
[11] Chow S.N. and Hale J.K., Methods of Bifurcation Theory, Springer-Verlag, New York,
1982.
[12] Christopher C. and Li Chengzhi, Limit Cycles of Differential Equations, Birkhauser,
Basel, 2007.
175
ë©z
[13] Coddington E.A. and Levinson N., Theory of Ordinary Differential Equations, McGraw
Hill, New York, 1955.
[14] Conway J.B., A Course in Functional Analysis, Graduate Texts in Math. 96. Springer-
Verlag, New York, 1985.
[15] Culver W.J., On the existence and uniqueness of the real logarithm of a matrix, Proc.
Amer. Math. Soc. 17 (1966), 1146–1151.
[16] ¶Ó;,o«£, ~©§,pÑ,®, 2005.
[17] Dufour J.P. and Zung N.T., Poisson Structures and Their Normal Forms, Progress in
Mathematics, Birkhauser, Basel, 2005.
[18] Dumortier F., Llibre J. and Artes J.C., Qualitative Theory of Planar Differential Systems,
Springer-Verlag, Berlin, 2006.
[19] Ecalle J., Introduction aux fonctions analysables et preuve constructive de la conjecture
du Dulac, Actualities Math., Hermann, Paris, 1992.
[20] Godunov A.N., Peano’s theorem in Banach space, Functional Analysis and its Applica-
tions, 9 (1975), 53–55.
[21] Goriely A., Integrability and Nonintegrability of Dynamical Systems, World Scientific,
New Jersey, 2001.
[22] Guckenheimer J. and Holmes P., Nonlinear Oscillations, Dynamical Systems, and Bifur-
cations of Vector Fields, Springer-Verlag, New York, 1983.
[23] Hajek P. and Johanis M., On Peano’s theorem in Banach space, J. Differential Equations,
249 (2010), 3342–3351.
[24] Hale J.K., Ordinary Differential Equations (2nd ed.), R. E. Krieger Pub. Co., Malabar,
1980.
[25] ¸jS, ÄåXÚ±Ï)©|nØ,ÆÑ,®, 2002.
[26] Hartman P., Ordinary Differential Equations (second edition), Birkhauser, Boston, 1982.
176
~©§
[27] Hilbert D., Mathematical problems, Bulletin of the American Mathematical Society 37
(2000), 407–436.
[28] Hill G.W., On the part of the motion of the lunar perigee which is a function of the mean
motions of the sun and moon, Acta Math. 8 (1886), 1–36.
[29] Hirsch M. and Smale S., Differential Equations, Dynamical Systems, and Linear Algebra,
Academic Press, San Diego, 1974.
[30] Ilyashenko, Yu. S., Finiteness Theorems for Limit Cycles, Transl. Math. Monograph, 94,
American Mathematical Society, Providence, 1991.
[31] Ilyashenko Yu.S. and Li Weigu, Nonlocal Bifurcations, American Mathematical Society,
Providence, 1999.
[32] Jost J., Dynamical Systems , Springer-Verlag, Berlin, 2005.
[33] Kelley W. and Peterson A., The Theory of Differential Equations Classical and Qualita-
tive, Pearson Education, INC., New Jersey, 2004.
[34] Li Jibin, Helbert’s 16th problem and bifurcations of planar polynomial vector fields, In-
ternat. J. Bifurcation and Chaos 13 (2003), 47–106.
[35] oÁ),ïI, 5ê,¥IÆEâÆÑ,Ü, 1989.
[36] o, 5.nØ9ÙA^,ÆÑ,®, 2000.
[37] Li Weigu, Llibre J. and Zhang Xiang, Extension of floquet’s theory to nonlinear periodic
differential systems and embedding diffeomorphisms in differential flows, American J.
Math. 124 (2002), 107–127.
[38] Llibre J. and Zhang Xiang, Rational first integrals in the Darboux theory of integrability
in Cn, Bulletin des Sciences Mathematiques 134 (2010) 189–195.
[39] 4G,oUQ, ²¡þ|eZ²;¯K,ÆÑ,®, 2010.
[40] Û½,Ü,Âr,ÄåXÚ½5nØ©|nØ,ÆÑ,®, 2001.
177
ë©z
[41] Lyapunov M., Probleme General de la Stabilte du Movement, Princeton Universtiy Press,
Princeton, 1947.
[42] Magnus W. and Walker S., Hill’s equation, Interscience Publishers, New York, 1966.
[43] Miller R.K. and Michel A.N., Ordinary Differential Equations, Academic Press, New
York, 1982
[44] Olver P.J., Applications of Lie Groups to Differential Equations, Springer, New York,
1998.
[45] Palis J.Jr. and de Melo W., Geometric Theory of Dynamical Systems An Introduction,
Springer-Verlag, New York, 1982.
[46] Pontryagin L. S., Ordianry Differential Equations, Elsevier, San Diego, 1962.
[47] Romanovski V.G. and Shafer D.S, The Center and Cyclicity Problems A Computational
Algebraic Approach, Birkhauser, Boston, 2009.
[48] Roussarie R., Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem,
Birkhauser, Basel, 1998.
[49] Rudin W., Functional Analysis (2nd ed.), McGraw-Hill, New York, 1991. .
[50] Smale S., Mathematical Problems for the Next Century, The Mathematical Intelligencer
20 (1998), No. 2, 7-15.
[51] Sagan H., Boundary and Eigenvalue Problems in Mathmatical Physics, John Wiley, New
York, 1963.
[52] Vanhaeche P., Integrable systems in the realm of Algebraic Geometry, Lect.Notes Math.
1638, Springer-Verlag, Heidelberg, 2001.
[53] p<, ±µ, Ágµ, Æt, ~©§ (1n), pÑ,
®, 2011.
[54] ÆM,H;, AϼêVØ,®ÆÑ,®, 2000.
178
~©§
[55] ÎR+,o], ~©§,pÑ,®, 2004.
[56] ^, ~©§ùÂ(1),<¬Ñ,®, 1982.
[57] ^, 4Ø,þ°ÆEâÑ,þ°, 1986.
[58] ^, õª©XÚ½5nØ,þ°ÆEâÑ,þ°, 1995.
[59] Üö! ±, ¼©ÛùÂ,þ!eþ,®ÆÑ,®, 2003.
[60] Ü<ö,¾,~©§AÛnØ©|¯K (1n),®ÆÑ,
®, 2002.
[61] Ü¥,¶Ó;,©G,ÂU, ©§½5nØ,ÆÑ,®, 1985.
[62] Ü¥,o«£,x²,o, þ|©nØÄ:,pÑ,
®, 1997.
[63] Zung N.T., Convergence versus integrability in Birkhoff normal form, Ann. Math. 161
(2005), 141–156.
179
¶c¢Ú (U©Ñ^S)
A
Q: Arzela–AscoliÚn
B
½©| BanachØÄ:½n
Bendixson½n Bernoulli§
Bessel§ Bessel¼ê
4; '½n
ؽ4 C©§
Cþ©l§ ½4
ØC¡ ØÄ:
ؽ: ؽ(:
C
Cauchy¯K CauchyS
~: ~K
~êC´ ~êC´úª
~©§ ~
ÐQ: мê
ÐÈ© Ð:
Ð(: Ð¥%
Щ: Щ^
Щ ЯK
Clairaut§ o:
3«m 35½n
D
êê êAÛ
êÿÀ êfª
üëÏ« ü Ý
180
~©§
üf ÝëY
³¡ ³
4íúª ½K
½ ÄåXÚ
Äþ ÄU
é¡/ª Dulac½n
Dulac¼ê õª©XÚ
E
©§ Euler§
Eulerò
F
uÑþ ¼©Û
£m £N
ÛÉÝ ©|
FloquetnØ Floquet5.
Floquetê EA
G
Gamma¼ê p©§
p5©§ pàg5©§
pàg5©§ Gronwallت
2 Gronwallت ÝE
ݧ .5X
2Â?ê ;!;
H
Hamilton¼ê HamiltonXÚ
¼êÕá î
Hilbert116¯K Hopf©|
·b
J
Jacobi1ª JacobiÝ
181
¶c¢Ú
Ä)| Ä)Ý
È© È©§
È©Ïf AÛê
4 4
ìC½ :
) (:
)35 )3«m
)òÿ )5
)éëêëY65 )éÐëY65
è )Ý!Ý)
)Û) )Û©§
Cq) JordanIO.
ÛÜ Lipschitz^ ÛÜI
ål ålm
Ý, Ý=
Ýéê Ýê
Ýê¼ê ýé
K
m«m
L
Legendreõª Legendre§
L’HospitalK ëY
ëY) ëY ê
") Lipschitz~ê
Lipschitz^ Liouville–Arnold½n
Liouvilleúª LiouvilleÈ
Lyapunov1 Lyapunov1
Lyapunov¼ê Lyapunov½
Lyapunovê
182
~©§
M
?ê ?ê)
N
ngàg¼ê _Ý
n~©§ nàg5©§|
nàg5©§| nþ
ngdÝ HamiltonXÚ
P
Peano½n ©§
©§ Picard½n
PicardÅg%C Poincare–Bendixson½n
Poincare©| Poincareè
PoincareN
Q
Û: Û)
T§ ê
R
?¿~ê Riccati§
f Hilbert116¯K
S
³U nت
¢) ÄgÈ©
ÄgÈ©35 Âñ
Âñ» Sturm'½n
T
TaylorÐm Aϼê
A¦ê Aõª
Aþ A
183
¶c¢Ú
Aê FÝ
ÏÈ© Ï)
Ó©| Ó;
òz(: ýÈ©
à«
W
ålm
È ©§
©§ ©§½5nØ
©§ÈnØ ©§|
½ ½4
½: ½(:
) Wronsky1ª
X
[: |
àg5§ àg5©§
5©§ 5Ã')
5') m
þ ":
ã Ç
"Ý
Y
Ø K Ø Nn
~©§ É©|
É; Âñ
k. ÏCþ
Û¼ê3½n `¼ê
`?ê k.48
184
~©§
Z
¥%
KÛ: I§
I ±Ï;
T ±Ï¼ê ±Ï
±Ï ±Ï)
gCþ fm
g£©§ 3«m
±Ï
185
;¶c¥=©éì (U3©¥Ñy^Sü)
©§ ~©§
Differential equations (DE) Ordinary differential equations (ODE)
©§ ©§
Partial differential equations (PDE) The order of differential equations
n~©§ ~©§
nth order ODE The first order ODE
) Ï)
Solution General solution
Jacobi1ª Щ^
Jacobi determinant Initial condition
Щ: Щ
Initial point Initial value
©§ ЯK
Second order DE Initial value problem
)35 )5
Existence of solutions Uniqueness of solutions
)òÿ È©
Continuation of solutions Integral curves
| )3«m
Line segment fields The interval of existence of solution
ëY 3«m
Continuous (Continuity) The maximal interval of existence
Ç
Tangent line Slope
Cauchy¯K 35½n
Cauchy problem The existence and uniqueness theorem
186
~©§
Picard½n Peano½n
Picard theorem Peano theorem
ÛÜ Lipschitz^ ëY ê
Local Lipschitz condition Continuous partial derivative
T§ ÏÈ©
Exact equation General integral
üëÏ« à«
Simply connected region Convex region
È©Ïf ¿^
Integrating factor Sufficient and necessary conditions
m«m é¡/ª
Open interval Symmetric form
5©§ àg5§
Linear differential equations Homogeneous linear equations
?¿~ê Bernoulli§
Arbitrary constant Bernoulli equation
Cþ©l§ ngàg¼ê
Equation of separation of variables Homogeneous function of order n
Riccati§ ÐÈ©
Riccati equation Elementary integral method
©§½5nØ ~êC´úª
Qualitative theory of DE V ariation of constants formula
~êC´ ©§|
Method of variation of constants System of differential equations
àg5©§ àg5©§|
Nonhomogeneous linear DE System of nonhomogeneous linear DEs
T ±Ï¼ê ±Ï)
Periodic function of period T Periodic solution
L’HospitalK Clairaut§
L′Hospital rule Clairaut equation
187
;¶c¥=©éì
Û) Ûª©§
Singular solution Implicit differential equations
gCþ ÏCþ
Independent variable Dependent variable
p©§ g£©§
Higher order DE Autonomous differential equations
ýÈ© мê
Elliptic integral Elementary function
PicardÅg%C ålm
Picard′s method of successive approximation Metric space
fm ýé
subspace Absolute value
Âñ 4
Convergent (convergence) Limit
CauchyS ålm
Cauchy sequence Complete metric space
Âñ Ø K
Uniformly convergent Contraction mapping
Ø Nn ØÄ:
Contraction mapping theorem Fixed point
BanachØÄ:½n ¼©Û
Banach fixed point theorem Functional analysis
Lipschitz~ê Arzela-AscoliÚn
Lipschitz constant Arzela−Ascoli Lemma
k. ÝëY
Uniformly bounded equicontinuous
k.48 ëY)
Bounded closed set Continuously differentiable solution
188
~©§
Eulerò ëY)
Euler polygons Continuous solution
È©§ )éÐëY65
Integral equation Continuity of solutions w.r.t. initial conditions
Cq) )éëêëY65
Approximate solution Continuity of solutions with respect to parameters
Ý= 5')
Transpose of matrix Linearly dependent solutions
5Ã') )éëê5
Linearly independent solutions Differentiability of solutions with respect to parameters
C©§ )Û©§
V ariational equation Analytic differential equation
)Û) þ
Analytic solution V ector
?ê `?ê
Power series Majorant series
`¼ê nþ
Majorant function n dimensional vector
Maximal value (maximum) Minimal value (minimum)
©§ÈnØ ©AÛ
Theory of integrability of DE Differential geometry
êAÛ êÿÀ
Algebraic geometry Algebraic topology
ÄgÈ© 4
First integral Limit cycle
ÄgÈ©35 Ý
Existence of first integrals Rank of matrix
189
;¶c¥=©éì
¼êÕá ³¡
Functionally independent Level surface
³ ØC¡
Level curve Invariant surface
: î
Intersection point Transversal intersection
ü Ý Û¼ê3½n
Unit matrix Implicit function theorem
È m
Complete integrable Phase space
ã ;!;
Phase portrait Orbit (tranjectory)
Hamilton¼ê HamiltonXÚ
Hamiltonian function Hamiltonian system
¼êFÝ ngdÝ HamiltonXÚ
Gradient of function Hamiltonian system of n degrees of freedom
åÆXÚ "Ý
Mechanic system Symplectic matrix
ÄU ³U
Kinetic energy Potential energy
.5X Äþ
Inertial system Momentum
LiouvilleÈ nàg5©§|
Liouvillean integrability System of nth order nonhomogeneous linear DEs
Liouville–Arnold½n nàg5©§|
Liouville−−Arnold theorem System of nth order homogeneous linear DEs
Gronwallت 2 Gronwallت
Gronwall inequality Generalized Gronwall inequality
Ä)| ÛÉÝ
Fundamental solutions Nonsingular matrix
190
~©§
Ä)Ý )Ý!Ý)
Fundamental matrix of solutions Matrix solution
Wronsky1ª Liouvilleúª
Wronsky determinant Liouvelle′s formula
Ý, êfª
Trace of matrix Algebraic complement
_Ý Euler§
Inverse matrix Euler equation
Ýê
Norm Matrix exponential
JordanIO. Ýê¼ê
Jordan canonical form Matrix exponential function
A Aþ
Eigenvalues Eigenvector
Aõª EA
Characteristic polynomial Complex eigenvalues
nت ¢)
Triangle inequality Real solution
(: ½(:
node Stable node
òz(: ؽ(:
Degenerate node Unstable node
Q: ¥%
Saddle Center
½: ؽ:
Stable focus Unstable focus
FloquetnØ Ýéê
Floquet′s theory Matrix logarithm
Floquet5. üf
Floquet normal form Monodromy operator
191
;¶c¥=©éì
A¦ê Aê
Characteristic multiplier Characteristic exponent
Floquetê ±Ï
Floquet exponent Minimal positive period
": '½n
Consecutive zeros Comparison theorem
Sturm'½n
Sturm comparison theorem Oscillation
~: Û:
Regular point Singular point, critical point, equilibrium
Âñ» 4íúª
Ratio of convergence Recursion formula
KÛ: 2Â?ê
Regular singularity, Regular singular point Generalized power series
I§ ÝE
Index equation Conjugate complex roots
Legendre§ Legendreõª
Legendre equation Legendre polynomial
Bessel§ Gamma¼ê
Bessel equation Gamma function
Q: ¥%
Saddle Center
Aϼê ݧ
Special function Conjugate equation
) ½
Stable solution Stability
Lyapunov½ ìC½
Lyapunov stable Asymptotically stable
") êê
Zero solution Algebraic multiplicity
192
~©§
AÛê Lyapunov1
Geometric multiplicity Second method of Lyapunov
½ ½K
Positive definite Negative definite
~ ~K
Nonnegative Nonpositive
Lyapunov¼ê Lyapunov1
Lyapunov function First method of Lyapunov
Lyapunovê ·b
Lyapunov exponent Chaos
ÄåXÚ ½4
Dynamical system Stable limit cycle
ؽ4 ½4
Unstable limit cycle Semistable limit cycle
±Ï; ±Ï
Periodic orbit Period annulus
õª©XÚ Hilbert116¯K
Polynomial differential system Hilbert′s 16th problem
Bendixson½n 4;
Bendixson theorem Closed orbit
Dulac½n Dulac¼ê
Dulac theorem Dulac function
Poincare–Bendixson½n PoincareN
Poincare−Bendixson annulus theorem Poincare map
è Poincareè
Section Poincare section
£m £N
Return time Return map
uÑþ ÛÜI
Divergence Local coordinates
193
;¶c¥=©éì
TaylorÐm ÐQ:
Taylor expansion Elementary saddle
Ð(: o:
Elementary node Strong focus
[: Ó;
Fine focus, weak focus Homoclinic orbit
É; ©|
Heteroclinic orbit Bifurcation
½©| Hopf©|
Semistable limit cycle bifurcation Hopf bifurcation
Poincare©| Ó©|
Poincare bifurcation Homoclinic bifurcation
f Hilbert116¯K É©|
Weaken Hilbert′s 16th problem Heteroclinic bifurcation
Q(©| .©|
saddle− node bifurcation Transcritical bifurcation
ú©| VÛ:
Pitchfork bifurcation Hyperbolic singularity
194