JV loekle ~

D,:..1 's 1'00 A\6£r6l'm ro

U)NW l~~

exts./- VY1At-e I~

~ :::ib+ 75 , o ~tf<h

~e c~e~ i ()'(CA c{ Qc5P CPvikJ #,e zf/4ri-/enl

oyJ_ cf 0rOlU 'rxki I J-he J.\1{s,'oo o} ~ -1).rJ. b.

'"@n:,t~id~ \n

TS /).. :::: 33 ard 6 = -1 I )l,e,n

'3~ = ~~.L-,+)

-:9 i = ~ {)liod is' =

1rlkcJe,< (). hu Me '1sro

CIA l\e rJ eAJ e)O

+he <½~~ c,._ hots

~-;; ciJ t\ -

(r) . ~ lwN c}hJ­

Jhe 'l5enwhf PY

C1Jrcl 1'.s ~ [(eJ odcl .

cJ tJn {~~,a,, A • n '"--Zt Cl ..(~f\J e _j

dJV,'sfoa iJ -4.

Eve~ <fJ~e'O

JS rJ ✓ di -< fc .2. . '{ A ,o ' I (Ji :; (3J) :: ~ j I _J,eal!e8 .mf ---rifJrtWfltHJ-i

o (fi)\..oo J\lrJJ ~ .Lt . ·

f J IA == 't J

{J.Q_ ~ @rt 1-,j : '.:' 4-{+ltz-tl

~ ~{i+i)+r :::c -4 . )Lt I , )_ eJt MeJ Me oe,rf1IU 1rrl Po' J

vok,r.i ~'-J,rJeol -~ -4, C&_'). $h,,N )-hbt} Jl,e szv'.a'l!l' cl a?J or/) chkge?f

ts of lAe ~~ g,)<f)

Ba J-i ,s i if) (1,ltg-oorJh rn / (A'/Y ~~'If

~ c~ c~ ~eokJle as one or- 11,e

J:ill'6' .hr6rns -'if I ltt, fl 1 -1j, fJ-1 -1 t +3 . U-e15e

..tiz+1 "'..J .tii +?:. ~ wrJ..

· 0t-t0-<= 16/1-23+1 = ,g ( ~~+iJ+-1 ~ ~hi.

§z+D:i..=- H{'+ ;y/" H =' 'ts (,!;l,/r3iJ+1 == gk+-1

011 en~ '2i5 ,C,s- ~11 A7 I .

sollVf00

!ltU~cJ ~ &tv,5100 cJ_rY,rY1 / e\leif

()... \S ti- Jhe ~

rs A~ '3ff Cd.. Ci\~) I

?lj,. 1

3j/f-I o< 3i + ~

:; 3i ( @r;;)+~) 3 3

- 1, { qrl+;)) , ,'s rm 1'~

'ff ~" ?it-I, {).(a2+::i.) - @i+O {C!itd-t;J 3 0

J-he ~ 5,, q k, qk+I, Q?) u1 ~+'8 .

s-oh.J,co

g{J J;\.ft1' 5,'on ~Jhm , evd ,oktJpt

!ft 1~ o[ Jl>e kro 39,- I 3j t-1 / o{J" 3J 1-~.

)GE ~tions

3 ().? - @i-t-()

- ~·-:ri3 -t ar;,zf.(t- 12, -t-'

- CJ { 3z3t 3 tR+ i) t- I

- qk+J

-rt ~-= 32+-~ t,. 3 - (!_2 t ;;f)3

z- ~';) i2 + 54z."f+ 36 f t- g

- 9 ( 31,3t ;;;;fR+ ~ff)t- g

-:;: q k+3

·(nc>€a.k5~ Lomy000 l>i\.11'.s'O)s' { (l)mJ

lkJi'n1>fo0 Jh, 10~= b 15 so.iii ,J,o be ~v,s1f:l-e

'a-- 0.'() <hJew-. @. 'f o , t'r> ~'cJrnb-o/1 rA / b ., , 'J' .Jl,erse

eXA:S},; ~ 1\,~e,,- c swJi J-w.. b =- M'. .

, l\'e vozs,'h? (). t b ~ 1'ocbWe ihfl,J 6 vv "ti Ob/--rd,\,,J1'b/e

'DeiHrutfoO ,;:: J fA ll~ 1 o= a'61:,I~ r 0~,

~ On IYJ ~~ d ts st,J'c/ ,!-o be a c,, o:,rooo

di\11's01> oF ~ orJ b ,·f boM d /a ad. d / b

~e} IA (),re( 6 pe z17ven ch~~/ v01'th

O\J- I eas~ cve of ~ero ob1 ~~ ~rn :z'ecm . ~

a"1Se!\,l-es} lo«rrnoo c/J.\J/So'!( oft ~ o,,J h I ~

':u ocd (l,l, h) ,:s fhe poS>'ts'\le th~i( ol ~otA:r~,~ Me thino,,JI~ .

@ cJ)Cl ard d/b & rr C /t?< ()_rd cl b Jhen C <!.d

<fh,e pa31h'Ve tJA\J ISt?crs of - ,~ a~

11 ~, ~ .2/1

b1

I~ vvhezsetls /-ho-<J.e o[ '30 Gl'a(:

11

~, 31

s,61

101 16, 30 . '!he po$ihv-e C,,=r, div1'sc=

tf - I ~ o-rr>l ao cas-z,e I 1 ~ J 3J 6 )' Bee.a~ 6 Js

,he 2-o.~e:rJr of J.hese (~ ,, occl {-1::i,3o)=-6 .

0cd c-'5, <S) = 5

o&J. c 15,)''1)-= J

gecl (-i, -36} = -1.

3 eel. (-1~, 3(0 = 6 = EI ;;i) ~ -t- '3o. l

~a1 c-g/ -3 0 =' -1 = t_gJ11 + c-3 6)<-D

Lk~Y\Jn·oo Cn,\Jen 1}.J,Je,,<, , o. C!vv/._ h, l')Od- boJI, o/

VoNli. ~ xew, Jhe~ 045) ,'n~ns -X prr/ /J-~ (o,b) ~ &n't hc'J

'f \"1'u th~e~ ()'l arr:/

Orse 'Ztffi) ~ scv.,J ~ /

6 , )')~ kM u/

be at> WNe{1 P251rr.e

vOhea:Aie-6 c)UJ( [ 41 b) "" j

E uc.LcleaYJ Pr~lhr0

Je/: rJ. o.rrJ. b 'he J-wo {0Jeer"s . §.. &.\J. (51 'oral /J~o>'Uwn

6\.= 0 h+tS', o~~<h-"'' /

'if ,~ h.ppen, JhJ- --n, '° o, »,en b)o. ffJ [~,b)~ b ·

whe0 75', f- v, J\J, ,Je b ~ ~, rfv pcfldw {)

t~e'2SS 't« 0,,,,J i(.:,_ Soth'!,~'ZT-b -=- ct;) --o) + cf~ / o f. "1(~ < tf,-

iJA €Jr1 vv {) s},o p . o M eovv {.s -e .

~ Jl-.;{S1oYJ f'"°LeJ>.s Co d-tn 1tW tt) urih'I ~Oto,e

:Zeot> 75emcu'rd~ apf€ao5 r ~ ~ Jl,e ((),-£iii s~ ~ vvhecre an-, is oL\uJ et) ~ er h·

1W

11,e c5'esull- lS -Me

'7vMoos_

{A-=- ~,b-tt!J / Ot::.t5j~b

b-=- Cf,;}, --t, + '?f ~ 1 0 Ccf~ C::.'"'61

o1

-;:::: lj.3

'lS".;i + "1)3 1 ° c.. "f3.c..~

o" n - ) -;: 'k on + tJ

%e ksJ- noo 7e7JD ?SeK«M~e-{ l)n ,·s Jh<:

rd_ [o.,b).

Ci) h·~ f J ( I o/37fl/ 3o5V 5olvth'oo

1a31~ ~ ~.305Lt+ 16~

'3 0 5~ - 1 ~,. J l, ;;; -J- 1 3 9

I b ~ - I I '3<i? -f ;;;1

1 g _ .3· 6 + O

;rel (I~ 3'H 1 3o')~ = 6.

t = ~4 - Jg

_ ~Lt _ (l3S- 5,~1,)

- 6 . } ~ ;;} - 'l- . 13]

-:c (, . }6;) - :f {3o51- )g.J1,i)

6 ~ ocd. [JQ",-:/-'tl 3os'D-= /;;J3'M:x'.f 3055';; / v0h01re

?( ~ , '3 Q CA,rrJ_ 0 =- _ 53'5 .

(?) h'rrl 3cJ C 1431 R~"JJ so,~1:>()

a 'o). 1 = I • 1Lt 3 t <E 1

I~ 3 = J ·~Lt -}- 51

~" ~ J,5l/ + [)5

5C/ ==- '9, ;;)5 t 1 ~ dJgJ¥ 85:::. d•i-t--1

~ - a.-:/ I -, CJ -:: J '7 t- ~ 1-- J,11± '3

~ - 1 3 + J

~

occl ( l'1?, ;;i;;i~ "' l

":/ == 3-~ t ) ~-== ,.~-tV

- 'd·1-~ ·1

-::: d· [a5-~-q) - :3 ,Cj

~ d-~5 - ·-;;-, q

- o}-olS- 1- . (5~-~·d5)

- 16-~S- 71.59

- } 6 . ~ 't - ol 3, 59

:: '] 6 . r. 4 - ~ 3 · { 1113 - I , g D "=' 2;q.tt1t - a3 • H-3

::: 3q. {~~1- I· IL/ 3) - ~3 -J',3

"' ?>'J· ~;;)-::/ - b-:;). 143

' :: ?,q. ;J.;)1 f {-6~)- )1t 3

l"" gcJ [d~1-, 1.1i3)::: :),;)1Xf l43cf

?l""'? q a nJ Q'" -6::l '

(_g) pnol ~cd ( 306/ 65i)

{;s':J.:: ,:!).306 -J- --4~

306-= 6.'tSf- 36

-45-= I ·36 + 1 3 {, ~ ~ ,q + 0

- 1 .11'5-1•3ni

~ <-;f . ( 6 Eit - c) · 3-o'i) - I · 3 ° t1

~ -::f. t~ - 15,306

vO~

q = r (6"571 3°~-:::. 657--X + ?,b6o- l!Oh~.

'X =: + a.rd O;; _ 15 .

0) lse J-he hlcL'rlello 1rtr11im ~ ohk'o "? 'X ~<J O ~~tl\c} ~ ~.

oeJ ( IM1 ~1-'£) = ll lf7' t @1~ ,

91~-:: a- )lq t 31/

I ) q -::: 3 , 921 t 11-

'3 Lt~~. 11 + o

NoV\) 1

11:::: Jlq- 3 ,32f

:;:_ Jlq - '3 · {o?1;;J - ~-n1)

~ (.J.1~/ 11V-= )JqXf- ~'111 '){ ~ 1- o.rv:I. cf -'3 .

b?(,-e75U1

Sf:L>

Ci) F;)'{} to{ ( ~:f;:)I 11! 11)

i-oh-emc1.

QI) Use JI., e &,J:Ji)&.D Ab-=' .lJm.:J Jo o bkuh

10~{%) ex o.rJ a s,Jn~~ J-he t,J/.v0,~ .

@ ~ul [561 -:f;;)J ~ 56'X+ r7J ® t11 ( :;i.Jr 1 1'3 i) ==- o1.trx + I 3~

@) tr [17671

o/3-ii)-==- J ;Jbq:;t+ .;;J3hJ,

'Jhe J;.,ea"6 U)opl,,onn'o~ e,gwJxn Cl?f + Jy "' C

ho..s o. so/,,J,on ,:r r,rrl or{j 1'[ d / C, WhP..re

cJ,,_. acJ (a/b) . TJ '.io,ju i's o'/) pir;i;r.,._la-<

soki.b«> of Jl...i:S 1VVJ.hon, '1hen a/I olhr+; <;0lviion5

?C~ -xo+ (J)-1c / a~ !4,- (iY ~n cn5ts> baa ~11° '

la'l1 \Js;j GAcL.'r/eP.o J.0-'Jf-n.r, oo

I -::f ~ a rrl_ ao

I ::I~=- ~ ·dO + 1;>_

f)o ~ I, 1;Q f- '8

) d~ I· g, 1--9 <g -;; EJ, 11

cr.J [/"1~,;;io):o -11

s,·Y"lU? -'j /,000 I G\ s-0/\Ah'oo +o J/.uj ,l~OO €'J£1J15.

~-== 1 :;)- g

;: I;;) - (ao- /·lf)

- ;;;. 1~-dO

=: ;J, (1 --;:J-'6·~-~n

~ ~. J-:,.;) - JS at£t 11-cX)

1 ~ a -11~ + ( -1-:JJ ·cXJ

M 1M hp~ JJ-,., s oe ltlb:n v() oh a 50.

)ODO :c .Lj ,RSO =: cVSo r~. /:J~ + (-19),;;i~

==- S'Do. /-;f-;z '-)- {-4J;JS';)Ro,

~= 500

?l-:: ~-+ (-5--)t- = 5oof (~) ccc 5oo+5t

Q~ ~-{!r) = -2,;;&J- [I!f) = -i.td5o-43/.

J. Solve Me hneP-t Dophlnhne eil/rJ.l,,op

d-4 X-t- }3~J--= Jg.

Sv}~

\Jsi] bt.clcl-eP.o ~vo on

d.1 atit( /?,F.

)3t = 6•;;;~ + }<Z

a 4 = I · ,, fl f b

}<!:== ?> · 6,

cf o1 C « 2r, 13 ~ - 6.

$-J'r11 p 6 /;rs 1 tA ~oli,,J-,·on <1-o Hv:s egvioJ-;oo (?'JClS) .

6 = ~L, .- ) '6

t ~ r;J4 - ( l3f?-15-~

6::: b•dlt-13'6

b ~ c •:)Lt 1 c:,) l3~

f"{)111Jh·~u >he cSelA.l,oo W ,'Jli ~

1 i -; 3 . 6 :" 3 r (, . ;;i4 i Ci) , 3R J J ~ ::- } tg -~ ~ -1- c- 'i) )3 (d

/ 'b c=c ;JLrx-,. 13 & / whe?5f'

~o-=- )<[ orJ i ::: -3

~e oJi>&cs ~l !Ah'oru

7L = -x,, + C-!t) :- 1 E-tf';!-) 1

~ ::: )~ + ;;3-l

~ ~ ~- (f }t" -3-(~)1: ~;:;- -3-4+·

~- stJlve k ~nea75 D>'0phanhne ez~ 5Ltx+ ~1cr ~ 906

solu}for? 5 t; z. , -1, Q ) f'

5~ -= ;? . ~ J +- I ~

~ I ~ I -l~ t- CJ

I~::: 1 ,9 -J- 3

q ~~,_3

8 ( ol ( 5lt1 ~ 'i) = 3.

'7,'rn e -:) q o b, o.. $u I Wi·= ,h, JP.is ,v,oJ,oo

ex,t:S} I

NovJ 1 3 -=-- I~- )·CJ

- l~ - (Q1-l~)

- ci·)~-~}

::: r;;). { s't- Q.~1)--;;; I

-;. @· 5 Lt - 15 · ~ I

M1,1 lh'p~ Jfif cSE~ )'OJ'J/,

qot"" 3. 30;;;, "' 30~ f .,;,. s1i - '5 ·JD 9 ob ~ 60-1 , 5 4 - J51 -c , ~ )

q o b ~ 5Lrx + :;i I} i,o hez,-,e,

~=- bolt ~ ~.:r --JsJo

?l ~ -x. + c ~ 1 t = 6 0 " + r~~ ) t ?f. ::::- 002.t + ---1 l

a ;c ~b - { ¼')f - -)5/b - ( ~)t

d ::- - )S}o - I <3 {

/,. chec~ i,ohe~e'Zf CA <;ro/tAhoo ~cS' Jhf D-0pl"'Uot/ne

'r,,oJ-/m 6-x+ 5~ :-: ;/~ e;x.,~} o-o noi.

5l--;;;.~,6f3

t ~ Q-8

td c~,I 1,J = 3

'Bw 3 does noJ J\J1•Je,:3 C¥:;), .5o fhe

q,"W:,·on ~-,s 'Dropl--..mhne -ezirAPioo Jie-s o...;

e~sl--.

E'--:x.e6CA~es

, Sol-ve Jt>e ~ 'opl--anh'ne ~~ans.

(t) 56xt ~dcJ == 1D '.

~J ~~1-x+ 3'5u = 11

f3) ,~~5u =4&

(A) } a 3x f- 3bo c} = °19

[5) }53''X - 5?J ::- -:J.

.0-cJ?5Ue r1ee

Jet n be I) J;~· 'PoSl}Ne <~ .

') wo t~e"5 (!', a,,/.. b = soil <1-v le.

C,o1'.\'1S1Awk mock.In n, ~,..,boUJd eJ. {)i ~ b c Yf)\?J 0.

1'f n o!A\m}eJ 1'-ie J~ tA-h, Jl,P.l- 15

Q---b := kYl ~ ~~ ( ~~/'.) k ~

'If n / {).-b ,J-/-,en a. rr 1,1i ,n U¥?Jwenl qr; .b

~ n , o.rd. ~1-e a j b ( rniJ_ n)

~ e~le ol-4 = 3 [,,-.-d. -=1)

':dnte- £)-4 -3-= ~ l ::;: 3, + -91~1/ (-n,cd-:t)

S{Y1Le -3l-ll= -LJ~~ -6,-:f-

131!1,k ~ 5 '-f f;;) Crnod i) 51'-nee.

dS-1 ~ ~ ) 3 = no~ o.. mu,J,ple of ;f-.

@ a= fA (mod;)

(€) r+ 0.= h {rnoln) ~ b :: ~ (n,oJV

@) T[ a~ b (rntil ti) arJ b;=c (rnal.11) J-heo a = dnnln)

@ rf tA -=. b ( 'YYJod n) o,ro/. e. = J &mJ nJ, cH,eo

c:A-tC ~ (bt d { rmJ;;J arc/ . rAC = txl (rnoJ ;;J '

( t ,) 'l r {~.. L (h)t( ,, ~) r} f)O'(J 1/i r} { • j; I { ( tn4I n) and. c1r ~ hr (n)O('' h ).

. . .

ao

I ,k A ,. (' vt · r J (r l(>r.)(J ri) dv?5 an;:3 ,

~s =: 3& = -1 (moel -1D ~5 ? - q (mocl·/.JJJ

~ 5)'' = (_ qf' ( mod -1}

:Jd.O = (_ ff {- {f (tnoo/ -'t 7) ()~

0 = (§ l) { g U ( rn:od .1t 7)

ez 1 -::: - J C tr>od ➔ ~ -

~I -RJ=- l(moel4i)

~o ':\ ;;J = 1 C m-od .JJ 1)

;;; - 1

@) ~ firJ .Jhe 1SerralndfJ?f'5 50 /6

Vl?hen SJ ar.d .LJ I

0he 00r0cu'~ v<Jhoo JJ° J\111tkci ~ rf i's 1 ·

Also -', / :: -I (md '2f-)

ed~ t.-dr;c'r<)Od v 65 ~

~ 1 = - J C mal .:; ~

'Jhe -~ndt% \A?h,e;() "-1 / 5 dtv, dPel

~) ttnl J-he aemcv(W?f vohen div ,·ciAh(J

) ~ t a J -t 3~ -H ! -+ · - · . -+ q 4 ! -+ 100 _, 1rJ 6d~no

We ~ve 1,~ o 6,~

4 ! ~ ~11-::: 0 (rocd ,~)

5! =: 5 · -'t ~ :r 5- o ~ o (rncx:I J:i)

6 '· = 6 ,5. l-, ! = 0 ( t-ooJ /,;/)

I

I ;o ~ = 0 (rocxl I~)

I.'+~~+ 3~+,.ln + -- . . +qc,!+1vo!

bO

Me

Ja.

_ J '. + ~ ! -t 3 ~ -1- O+- · · + 0 (tr,ocl 1V ~ I +dt 6 c:rooJ I~)

-: 0) { rntd. 1 iV ~

15 q.

1 /~ --/.

;jV'ro

l'ow.'6 ~. 0he vh'{)e!).'15 e,,.,'fj'lfl,let\.W a-x 9 b ( mriJ r,)

~ 1)- suliJiw ,'f ()Ir./. ~ 1'.P ol / b 'IOh~

d = 30,1 {o.,Q . rt d/b . Hie,') 1'1-. /,'(,I) of

~ Li f rt ton8m€NVI- s-o /uho,u 'mo Mo I? .

'Jh e Co?j<IUe-N.P 15 ~ wVPtleMJ- +v -Yi e

J;.heg.75: ~ Vb pl-wnfrne ~h£))-ro0 ~ ·

O'X-b ~ r,~

()._~- n0 ::: b_

1li1'.J ~ Vll)l}Y0() 'Vt,,-, ~ "yo 1veJ I '{ d. A. 11,,e oJtiero ~~ tX7£-

?1-= ?to t ~ t­

a~~ -+¾ t CO s-dve Jhe ~h-eonf ~r,eHLL~ 1 gx-= ?Jo (ma/4~)

<5clIA-hV()

'f\~1) I

) gx -=- go ('rYWI 10

Jfsx - 30 = ..L/'11 J8?(-1tdQ -;;: 20

~&= a.1Ef6

)t ~ '3·6.

~, B'cel (11:1 "1 :D= £

g,'yt2_e {, J\1Ad~ ?,o) ~ e~ (}.. So'L.bob I

Jb 1'm ~ ()r't(> fsolvcheo 1S :t = -4.

<Tl.1e $1?( So1~ v~ -

'X? -1, + (-1tf·)t

Cj'-x ::c ;;i I Cm-d 3Q)

9?< - ~ I == ~ 0c} qx_ -- 3DlJ ::- ~ I.

A-l~ ~o-= 2, .q+ -3

G) -;:: 3 · 3.

rf J { 'f, 10 = '3.

Sl·n~ 2> ~"fclej a 1

l~ :3 = ~ o --- ~ ~ q.

~ I -=- 7- 3 -: 9 f ~0 -

3 . qJ

We ~ve '9).::::- Ol?<-?>o(f,

~ 1 '° 9 c-~LJ- 30 {-V ,> . 'Xe;;; -;;? 1 J ~ = - ;J-.

1he Mi~ Sbl~m a?Je ,

-x= -~, + (~)t -X= -;;:,>J~JOi· '7 t.:::01 J1o<

-x = . - :;i 1 , - n I - 1 r moc1. 3;;).

& ) S,0 Ive Jtie J·nems Co nu c>U&hl~ a 5X = 15 { rrd ciYI)

so1uhoo as-x ~ 15 { rrxxiav

as-x :-' 5:: d?J g c)S?f - at := I:;·

Abv ~q --=: J--as+1

Q5== 6,11 + 1

/4 ~ , ·11.

crw' ( 61'5, aV ~ I

b ) ~ cQS- t ·f ~ ~'S - 6 r ~q _, .:;is-J = d · ~':5 - b . ;?q

We h!!ve ci5?t - ol°ltJ := 15'" , m1,,1i}fp~ ½J 15

I 5 " I 5 f -;J . 'J<; - 6 ::1 ;J

. · . ?t. = 1 o,s 0i nd ~ =- q 0

{j"he ~-o1v.JmM 6\<oe

-x ~ 1o5 + ( 9f-} l , -1: po

?f_,;;:: Jo'5+ cQ9i , + ✓ o

~?( ~rseJ

· -Sdve .J-1,e fv 11.ow,rcJ };,'()€{Ko -lo~oUerv'eJ.

Ci) 5-x :s ;;J Cm¢ '()tJ ~) (:/X = I 5 ('tflvd qL}

Cf) 3 6)( -;; ZS ( r-rru:i lo~

(4) 3.lf'X :: 60 (rrnd qg)

{5) I .Lio l' = I '33 { n,al 3ol)

. Necuvenle Kelahoo

The geneal rs Ovdes linears hwmeg eruons oecussenle selabop

Consant Coelbuerts

Onm a An, nzo, whee d ia a

s Conant . SheeOnh odepends ord on

(mme elbte psedecessb he Delahion s Saud to

iast Ordes be

The unih ue Solubon of he

Te ssenee Selahon ant d4rn. n>o, d a

Consart wlh thuhl Condi'hion A A s

an Ad n>o .

C Solve an=_Ad"

the Tecuosente Telahon

498 whese na arod

Solukon The Solusen

Herve d- 7) Un = ao 7

qen 02 8

An 7", neo

Sohe Soluboo

Unt :5 An

d 5

olubor an = 5)D

Sohe -44n0, nO, a, - 5

Solwhon

d:

SDubon

Crive 5

A(-5

an ()(

Solwbon of he deuSehte G) ird dhe Unngne

Gelaion 6 0-7Ao-10, n a= 343

Sulusop 74n-1

d %

s Solubor naZ)

miser 343 3

-343

343 x6

34

An= a16

s) hind 2if Aot 5 4, ano Jos no and a-2

Sdtuho Is uinead The Telahoo oetusoene

ket bn= n, hen dhe Selahon betaroes

_ is bn ,5

The Solhoo b= 5

de25)

Ola 3), a50

(9 Sove he selahor

ond Al.

Soluho Thssehhoo 's a decussenee

eahon wh Vaoobble wethuenk

Og= -4,: a.l = a!

3.4 3 al 3

=4 = 4.32) : 4

teuses Tenbons sole dHe TLUTSEnLe

(a)4ao- 5an- n

( 20n- 34 0, n2', 4 -81

Seond Osdes ibens Horernous

Keurosene Peloon Wih onsdavnt loefuiènds ewosen

he genusul doar of Setorel

Oces homogeneons oelahon

wih Conslant Coelhiuen is

anke an= C&"

Co cC ct S cs"D n-

(Co+C4 C)ca" =

C+ Cs+ Ca

he Ct G+ G-0 Called egnahorn

dhe chsaekeseshc egnahon het and a

be he so ofhe egvahon.

Case (A Distint Teal

Df aro ae he dskn Teal

he Soluboo

an 4) + ()

) Seke ant n- 6On-a 0, na, -I, a, 8.

he Seeusoenie elahon

5oluhoo 1 an c*"

n- Gn- = Cs

On-a = C3

E gabon belomes

C +*-6 )-d 2 T+T-6 = O

+3 (-a)-d

-, .

ane,a)+G (3

= G+ 6 ( 2L,-3% =8

C+- C3a =8

--O

- C

-(9

5D a -3 = 8

3)T 4) 5 -10

(1=>C-2--

C,

On a-2 (3":

) Solve he SEuoente delahon

aan74h-5n3 a , 4, = 5

Sduho 1f an C?

an- C n-

An-a C

n-

Egvahon becomes

ac 7 c-3c

7st3]- C

-F6+3 =d

t49- 24

7f52

n C + ()"

Ao C, (s)+G (%)= c,+G:2,

T 4-5, a,= + (4)- 3Gt-5

-cD

3G+4=5 -

G2 6C, +a 10 -(

C3)-CD=> 50,=8

an (% )Svve the selabon ntFpt Fn. oeusSenie

Wheoe nao, and o0, )

Soluhon n

nt Fnti=Cs

nta, Dta Cxta,

Egabor belormesS

Ct c

e [--i)- 2

+4

Fo G + = 0 -<1)

F(1)4 G()=| - ()

Dts)> ,0tvs)+ Glts)= o -(3D

caD a G i+v6)+ G0-)= 2-

()-44) -> (aVs)- -

CaVs

F -

Kepeaed Ken oots

c)le he Jeussente ebbtn

ana44p+ 40n, na0, , a,-3

Ssohuboo LAn C" nt

ant- C

nt2 Ont C

An )+GK)

-D

G)ove he SeuTsende elabr

a6Ar+9na O, n2R, a=3, , = 1Q

Sohukoo nt

An= C, an- C, an (o

6+ 9=0

-3-o

33

anC,)+%n (3) C,5 5-CD

G+ Gl3)= la.-A)

5+ = IR

3 -3

An 5 (")- n{#")

Coroplex Koots

Sdve he seursente oelahon

an= 0An-a aao=\, a

Solubon n-

An=C C*', an-Cs

C ( C3" C%")

c S- a +)=o

t

an C(t)+ G (-

aC, C4D+ %l1-)= 3-e)

+i)=>

-(D-> ((-2i)=(-i)

S(i+)

1-

Etesaises

Splve he SetwssenLe elbons

)an= 54p-t 64n-2, n22, =l, a, 3

(a) a4n4-4ntt+54, 0, na0, -, 4,:-8

(On+tn-0, n20, d-0, 3

(A) an + n- t 4n-aO, n2a, 4 = 1 -3.

elakoo Non homog entouS Seussenie

Corsile he hwmoneLOUS

hiss oles selakon ,t G, n-= ks, whorsek

Conshav Ond nE z. 1F " is o a Soluio

he assouaked omogeneons elahon

nt G 0 . hen a A" hese A i's a

Consarnt T 1s a Soluhion of e lossespondin�

homoneneos seakon ,

han an= B»3", B so lonhn

onsides he homoge neons non

Seond odes seahoo ant Cdo-t G k whese k is a Condank

pf " s a Sobhon o he hormggencou

Telabon An As

( an whese

a B wheae B a Consland

)C", C a a lonkond

P) he sowo

0Solue he 0 Cwssenie eabon

n-3 An-5 (7"), nai, ad

Solubon an 34p-

Ch)

Sinte f- 5(), dake A (4) A

Epvabon A()-3 (Al)) = 54 be lomes

A T-3)- 577 4A 35

35 A

a29) an Gn + An

An

nive Ao= .

()+ 25(4)

ot =- - 27

3(9)+()

Sole he seuente bho

4p3Ao = 5 (3'), nl, -2

SD/uhon

an= 3An-1

Ch )

nye f 5 (). nke Bn () a Bo- (")

tgvabvo betoroes Bn6)-3 B(-D(3)- 5(8")

P 3 Bn3- 2Bn-D)= 533

3B(n-t1) = 15

B 5

a 5n )

an

4C(s*)+ 5n (")

Uivern

a (3)+o =

an (:")+ 5n ()

4 5n)(3)

G)(E)he he oeahon eusenie

nta+ 3@nH = 3, nao, a 0, a, |

Scduboo ke ay = (be ke Solukoo of he

homeqenons e9 nabop

DC)-o

h A GD+ B(-2)

( D)

ine f (H= 3", doke

D/2+ 2 D")+ & D3)= a"

20D

D

a (3)

A D+ B (-3+(

4)Sabve Hre DecudseniO Tehhon

ana 8ap+ 64, = 85)4 6(4'), n> o

- la, a, =5

Souhon bel ap - C be he Solubon ol he e

egvako» Santtl6a,o, Iheo

hormonencovs Bnto

ad

Cxt

-96t16 =O

aG")+ AG')+ Bnu) dake

Hee ft)=s5)+ 64D,

a CP c(5D+ Dil4") b=c5"*"')+ D(o+iðG")Ant

Onta

nbshdnhva eohdn dhe

c 4D (nr: (4*) -sc6Dow)

+6c )+ DHG)) = 3(5)+ 6/4 Compong Coehuers, of 5°) and 4

asct 4o H6c= s5°)

asc- 4oC+ 16c = 8

C 8

8Dt16-3o De) +16DJ- 6 A)

D l6n+64D+ 64 - 3a-&tn-32% nJ= 6

DC3) =6

p

an A 4+ Bn 4)+ s$)> 4)

niveo a A+%=

A 4

Coiven a,

a, A +4B+40t 3 5

48+ 56t 5 4B -51t -207

B=a07 16

apn4 (4") - 2o2 n(4) + s (5)+2 i4) 6 6

selahon E)Slue he SeUToenie

Ontot4n+44n= 7, n ,=1,4-R,

olnboo of he Ael an= c" be he Soluhon

homoneneos egvahonD

C +45+4)-

+= o

Ch

Sinte th-7, ake a éa C

On AC+ Bn(-D+ 4 an

CSelve Aan4,t +34, = -a00, n>o A ant ven an+2

ao 3000, a= 3300.

eoluhon n c be he Sobon of he

selabon egtothror Anta 14nt t 34,=o

homeneneow

Apta = CxTKR Ch Ch) C

nt 3C

c -43+3) -0

3,

Hene C, ()+6")

ne d(n) -a00 = -200() i Svluhoo

of ho homoqeneou ghoken selakbor take

a An s Conshant A Some

A (n+a)- 4A(nh)+ 3An -00

A = -00

A = 100

Hene anC,(3)++ lo0n

Qo C+( = 3coo

3C+tl00 = 3300

-) 3,+( = 320

C)-(D=2 aC, = 206

3e00 /00 I00

an )0o(3)f A00 t 70on

Ezesises

Sohie he seussente delaho

)On4an ant>, 0, =/

2

) Roe-an 3n-n, n20 Ao 3

(3) ant aan 5 ,h0, ao =

a,n0, a= 1. A) t

$)Gt ara 64n+, t 14h 3)+7(a") 3)+ 73")