eng-math-3, ch-1, 1st order ode ,
TRANSCRIPT
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
1/94
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P , ]ceb 3
YC]]DHBN-BFEHFBBWHFE-
@C^AB@C^HL\-HHH, LA-
3 HF G@B, JHW\^-KWNBW-KWNHFCWX-
NHJJBWBF^HCD-
BSRC^HKF\9 pcrt-3PYLA-3? JHW\^-KWNBW-KWNHFCWX-
NHJJBWBF^HCD-BSRC^HKF\9P YCNZCFLBN-
BFEHFBBWHFE-@C^AB@C^HL\, 5ta-BNH^HKF ,
GX BWUHF-IWBX\_HE P
Y^K]HL\ CWB--6 3.3 Gcshl lkflbpt cfn hnbcs, 3.7 sbpcrcgdb nhjjbrbfthcd bqucthkfs,
3.< `knbdhfe? sbpcrcgdb bqucthkfs, 3.> rbnulthkf tk sbpcrcgdb jkr`YKpthkfcdP, 3.2bxclt-nhjjbrbfthcd bqucthkfs,3.: hftbercthfe-JcltkrsYH.JP, 3.5 dhfbcr-nhjjbrbfthcd-
bqucthkfs, 3.1 @knbdhfe? Bdbltrhl-lhrluhts, 3.8 Krtakekfcd-^rcobltkrhbs kj lurvbs,
3.3= cpprkxh`ctb-skduthkfs? nhrblthkf jhbdns, Htbrcthkf9 3.33 bxhstbflb &
ufhqubfbss kj skduthkfs, GKKI? Cnvcflbn-bfehfbbrhfe-`ctab`cthls,5ta bnhthkf
gy Brwhf-Irbyszhe P
@y gdke jkr Ghk-`bnhlcd Bfe lkursbs ? -6 www.`bnhlcd-h`ceb-
prklbsshfe.gdkespkt.lk`
7=37-
7=3
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
2/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 7-KJ-1=
-----------------------------
^abrb crb fk skurlbs hf tab lurrbft nklu`bft.
^cgdb kj LkftbftsY3.3P \^CW^ ABWB9 ......................................................................................................................................... >
EBFBWCD-\KDR^HKF?- ...................................................................................................................... >
]CW^HLRDCW-\KDR^HKF?- ................................................................................................................. >
S 3 ) stctb tab krnbr kj nhjjbrbfthcd bqucthkf cfn vbrhjy tact tab ehvbf juflthkf hs c skduthkf ; ........... 2
S> ) stctb tab krnbr kj nhjjbrbfthcd bq cfn vbrhjy tact tab ehvbf juflthkf hs c skduthkf ; ........................ 5
HFH^HCD-ZCDRB-]WKGDB@?- .............................................................................................................. 1
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
3/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb
\B]CWCGDB-BSRC^HKF?- ................................................................................................................ 3>
@B^AKN-^K-\KDZB-\B]CWCGDB-BSRC^HKF\ ?- .......................................................................... 32
S : ) jhfn c ebfbrcd skduthkf , labli ykur cfswbr gy sugsthtuthkf ; ....................................................... 3:
S < ) jhfn c ebfbrcd skduthkf , labli ykur cfswbr gy sugsthtuthkf; ...................................................... 35
S 7< ) skdvb tab jkddkwhfe hfhthcd vcdub prkgdb`, cfn labli ykur cfswbr; ............................................ 38
HFH^HCD-ZCDRB-]WKGDB@ YH.Z.]P? .................................................................................................. 38
Y 3.> P stcrt abrb .......................................................................................................................................... 72
BSRC^HKF KJ ^AB JKW@ ?- ........................................................................................................... 72
AB@KEBFBKR\-BSRC^HKF?- ........................................................................................................ 72
S < ) jhfn tab ebfbrcd skduthkf kj jkddkwhfe bq ; ................................................................................ 7:
BQ # 3.2 Y \^CW^-ABWB P ............................................................................................................................. 7
Y 3.: P stcrt abrb .......................................................................................................................................... :=
HF^BEWC^HFE-JCL^KW\ YH.J 0 J(x,y) P?-.................................................................................................. :=
Bxc`pdb # 3 ?- ...................................................................................................................................... :3
Bxc`pdb # 7 ?- ...................................................................................................................................... :>
R\BN-KJ-HF^BEWC^HFE-JCL^KW\ YH.JP?- ....................................................................................... :5
@B^AKN JKW JHFNHFE HF^BEWC^HFE-JCL^KW\ YHJP ?-............................................................. :5
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
4/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >-KJ-1=
Y3.3P \^CW^ ABWB9
Bxpdhlht-skduthkf06 Yy0j(x)P9
H`pdhlht-skduthkf06Yx7+ y
70 l P9
EBFBWCD-\KDR^HKF?-Uabf tab fu`gbr kj Crghtcry-lkfstcft *l+ hf
tab skduthkf hs bqucd tk tab krnbr kj tab nhjjbrbfthcd-bqucthkf9
Bxc`pdb?-
Y-6 y 0 tcf Yx+lP P9
Uabrb , y hs tab ebfbrcd-skduthkf kj tab jkddkwhfe
nhjjbrbfthcd bqucthkfs ? Y8yy+ yx 0 = P9
------------------------
]CW^HLRDCW-\KDR^HKF?-Uabf wb csshef c ]crthludcr-vcdub kj tab Crghtrcry-
lkfstcft *l+ hf tab ebfbrcd-skduthkf tabf tahs typb kjskduthkf hs lcddbn ]crthludcr-skduthkf9
Bxc`pdb?
Ebfbrcd skduthkf 06 y 0 tcfY x + l P 9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
5/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 2-KJ-1=
\k csshefhfe c vcdub kj Crghtrcry-lkfstcft * l 0 7 + 9 abf, kur cgkvb ebfbrcd skduthkf gblk`bs, c pcrthludcrskduthkf tact hs 06 y 0 tcf Y x + 7 P 9 wabrb Y l 0 7 P9
--------------jhfhsabn -------
S 3 ) stctb tab krnbr kj nhjjbrbfthcd bqucthkf cfn vbrhjy
tact tab ehvbf juflthkf hs c skduthkf ;
Ehvbf ?-
Y P9Y P9\kduthkf?
Y
P9
Y P9Y P9Gy ushfe nhjjbrbfthcd kf gkta shnb w.r.t
x kf cgkvb bq-G.
Y P9Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
6/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb :-KJ-1=
.
.
Y P9Fkw, gy pdclhfe tab vcdub kj bq-L cfn G hf bq- C, tabf bq-C
gblk`bs,
Y
P9
Y P9Y P9Y P9
.
Y -6 1x 0 1x P9
D.A.\ 0 W.A.\
Acflbn prkvbn
LKFLDR\HKF ?-^ab ehvbf nhjjbrbfthcd bqucthkfs hs tab 3
stkrnbr kj
nhjjbrbfthcd-bqucthkf, cfn cdsk tab ehvbf juflthkf Yy0 1x>yP
hs c skduthkf.
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
7/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 5-KJ-1=
----------jhfhsabn abrb ------------
S> ) stctb tab krnbr kj nhjjbrbfthcd bq cfn vbrhjy tact tab
ehvbf juflthkf hs c skduthkf ;
Ehvbf ?-
Y-6 y
0 : P9
Uabrb Y-6 y 0 xP
S 3< ) skdvb tab jkddkwhfe hfhthcd vcdub prkgdb` ;
Ehvbf? -6 Y P9Hfhthcd-vcdub-prkgdb` Y H.Z.] P?
Y P9\KDR^HKF?Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
35/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
36/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
37/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
38/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb JHFH\ABN ABWB
------------------------------
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
39/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
40/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >=-KJ-1=
tab skduthkf tk tab bqucthkf Y P9 hs ehvbf H`pdhlhtdy gy {J(x,y)0 L },Cdtbrfcthvbdy stcrthfe whta {
}, ehvbs tab skduthkfH`pdhlhty gy jhrst hftbercthfe w.r.t y 9
--------------------------
BQCL^-NHJJBWBF^HCD-BSRC^HKF?-Nbrhvcthkf?- wb rb`b`gbr jrk` lcdludus tact hj juflthkf
R(x,y) hs lkfstcft cfn hj ht acs lkfthfukus ]crthcd-
nbrhvcthvbs ,tabf ht tktcd kr bxclt nhjjbrbfthcd hs , tact hs ,
Y P9\k,{?. -6hj , }9\k, bq-C , gblk`bs ,
Y P9Y P9YP9{?. nR(x,y)0= }9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
41/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >3-KJ-1=
Gy lk`pcrhfe bq-G & bq-L , tabf ,
Y P9Rshfe pcrthcd nhjjbrbfthcd kf gkta shnb, sk,Y P9Y
P9
Y P9Cdsk ,
Y P9
Y P9Abflb,Y
P9Abflb , bq C,G cfn L crb bxclt ,
-----------------jhfhsabn --------------
Y 3.2 P
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
42/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >7-KJ-1=
S 5 ) ehvbf {R(x,y)} , jhfn tab bxclt nhjjbrbfthcd
bqucthkf Y n(u)0= P , wact ifhn kj lurvbs crb tab skduthkflurvb Y R(x,y)0lkfstcft P ;
EHZBF ? Y R(x,y)0df(x7.y7) P 9\kduthkf ?-
\kduthkf ?-
Y P9Y R(x,y)0lkfstcft, n{R(x,y)}0= P9 JKW-NB^BW@HFHFE-^AB-ZCDRB-KJ-BQCL^-NHJJBWBF^HCD-
BSRC^HKF?-
JKW@RDC?-
Y P9Hts Kur bxclt nhjjbrbfthcd bqucthkf .
JKW-NB^BW@HFB-^AB-ZCDRB-KJ-@(x,y)-CFN-F(x,y)?-JKW@RDC?-
Y P9Lk`pcrb bq C & G , tabf , wabrb,
Yn{R(x,y)}0=P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
43/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
44/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >>-KJ-1=
S 72 ) c. ) Crb jkddkwhfe bqucthkf hs bxclt ;
g. ) skdvb tab hfhthcd vcdub prkgdb` ;
Ehvbf ?-6 Y P9Hfhthcd vcdub prkgdb` YH.Z.]P ?-
Y
P9
\kduthkf ?-
Y P9..
Y P9Fkw, sk,
Y P9Y
P9
JKW-NB^BW@HFBHFE-^AC^-BSRC^HKF-C-H\-BQCL^-KW-FK^?-
Jkr`udc?-
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
45/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >2-KJ-1=
Y
P9
]dclhfe tab vcdub kj bq-h & hh , hf bq-C, tabf bq-C gblk`bs,
Y P9Y P9 {?. bYy/xP0b }
Y-3 0 +3P9 ybs bq-C , hs bxclt ,
JKW-\KDZHFE-^AB-HFH^HCD-ZCDRB-]WKGDB@?-Y P9R\HFE \B]CWCGDB-@B^AKN?-
\k,
Y P9Y P9Y
P9 hts kur sbpcrcgdb jkr` kj bq-C,
Fkw,
JKW-y ?-
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
46/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >:-KJ-1=
Y
P9
Rshfe hftbercthkf kf gkta shnb w.r.t
Y P9Y P9Y
P9
Y P9Y P9 {?. -6 bl0l3} 9Y P9Y
P9
JKW-l3?-
------------
Hfhthcd vcdub prkgdb` YH.Z.]P ?-
Y P9------------
Uabf Yx0-7 , cfn y0-7 P ?-
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
47/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >5-KJ-1=
Gy pdclhfe tab vcdub kj x cfn y hf bq-_ , tabf
Y P9Y P9Y P9Y P9Fkw, pdclhfe tab vcdub kj YP hf bq-_, tabf bq-_ , gblk`bs,Y P9Y P9Y P9 hts lkrrblt-Cfswbr, Xbs, ekkn ,---------------jhfhsabn---------
S 32 ) \akw tact tab jkddkwhfe nhjjbrbfthcd bqucthkf hs bxclt
cfn skdvb tab` ;
EHZBF?-6 Y P9\KDR^HKF?
Y P9JKW@RDC?Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
48/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >1-KJ-1=
Y
P9
\k,
Dbt, Y P9 ,Y P9\k, gy ushfe pcrthcd nhjjbrbfthcd kf bq-h & hh ,
Y P9Y P9Cdsk,
Y P9Y P9Cdsk, abflb prkvbn tact,
Y P9\k, fkw, wb ifkw tact ehvbf nhjjbrbfthcd bqucthkf-C, hs bxclt,
abflb, tab ehvbf nhjjbrbfthcd bqucthkf-C, hs bxclt, sk ht bxhst cjuflthkf, Y J(x,y)P,
^abf,
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
49/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb >8-KJ-1=
Y
P9
Y P9 LKF\HNBW-BSRC^HKF-L ?-Y P9Gy ushfe hftbercthkf hf bqucthkf-L, w.r.t YrP 9
Y P9Y P9Y
P9
^ahs bq-B , hs kur stcfncrn bqucthkf,
{?. }\k, fkw, wb oust wcft tk jhfn kut tab vcdub kj YP tabf,wb put tab vcdub kj YP ct kur stcfncrn bqucthkf-B, tabfwb ebt, sk,
JKW-ZCDRB-KJ-YP ?-Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
50/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 2=-KJ-1=
Gy ushfe tab pcrthcd nhjjbrbfthcd kf gkta shnb w.r.t Y
P,kf bq-
B 9
^abf,
Y P9Y
P9
Y P9{?. }9{?. }9Y P9Y P9
Y P9\K, wb lcusb tk hftberctbn Y P jkr ebtthfe tabvcdub kj YP w.r.t YP9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
51/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 23-KJ-1=
Y
P9
Y P9Y P9Y P9Fkw, gy pdclhfe tab vcdub kj Y P hf tk tab bq-B ,tabf 9
Y P9Y P9Y P9.
Y P9Cfswbr.
-----------------jhfhsabn abrb ----------
S 35 ) sakw tab jkddkwhfe nhjjbrbfthcd bqucthkf crb bxclt, cfn
skdvb tab` ;
EHZBF ?-6
YP9\kduthkf?-
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
52/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 27-KJ-1=
Y
P9
Y P9Y P9{?. }9 JKW @(x,y) CFN F(x,y) ?-Y
P9
Y P9\K,
JKW-JHFNHFE-^AC^-BSRC^HKF-C-H\-BQCL^-KW-FK^ ?-Y
P9
Y P9\k,
Y P9.
Y P9Cdsk,
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
53/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 2
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
54/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 2>-KJ-1=
Y
P9
Y P9Hts kur stcfncrn bqucthkf,
Fkw, wb jhfn tab vcdub kj e(y) cfn ttabf wb pdclbn ht hf bq-B,
Uabrb ,
{?. e(y)0lkfstcft kj hftbercthkfs }9
Y P9Rshfe pcrthcd nhjjbrbfthcd kf bq-B, w.r.t y 9
Y P9Y P9Y P9.
Y
P9
Rshfe hftbercthkf w.r.t y .
Y P9Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
55/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 22-KJ-1=
Y
P9
]dclhfe Y P hf bq-B, tabf ,Y P9Y P9Y P9{?. Y P }9Y P9
.
Y P9..Cfswbr,Hts c lkrrblt cfswbr , Xbs,
----------------jhfhsabn--------
\tcrt jrk` S 38 ) , bx # 3.2 ,
---------------------
S 38 ) sakw tact tab jkddkwhfe nhjjbrbfthcd bqucthkf crb
bxclt cfn skdvb tab` ;
Ehvbf ?
Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
56/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 2:-KJ-1=
\kduthkf?
Y P9Jkr`udc?Y P9 JKW-ZCDRB\-KJ-@ & F ?-
Y P9Y P9 JKW-NB^BW@HFHFE-^AC^-YBS-CP-H\-BQCL^-KW-FK^?-\k,
Y P9\k,
Y P9Y P9Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
57/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 25-KJ-1=
Y
P9
Cdsk,
Y P9Y P9Y P9Y P9\k , abflb?
Y
P9 abflb prkvbn tact
bq-C hs bxclt nhjjbrbfthcd bqucthkfs ,
Jkr \kdvhfe Cgkvb Nhjjbrbfthcd Bqucthkfs ?Ub ifkw tact bq-C hs bxclt nhjjbrbfthcd bqucthkf tabf, hts
`bcf tact ht bxhst c juflthkf , tact hs J(x,y) 9
Y
P9
LKF\HNBW-YBS-hP ?-
Y P9R\HFE HF^BEWC^HKF\ UH^A WB\]BL^ ^K *w.r.t + x .
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
58/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 21-KJ-1=
Y P9Y P9{?. -6 e(y) 0 lkfstcft kj hftbercthkf }9
Y P9 Hts kur stcfncrnbqucthkf9
\k, wb nbtbr`hfbn tab vcdub kj e(y) & tabf pdclbn hts vcdub hf
bqucthkf-B ,
JKW-ZCDRB-KJ-e(y) ?-Y P9Y P9Y P9
Y P9 hts kur stcfncrnbqucthkf .
JKW - e(y) ?-Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
59/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb 28-KJ-1=
Fkw , gy Rshfe ]crthcd-nhjjbrbfthcd kf gkta shnb w.r.t (y) kf
bq-B 9
Y P9Y P9Y P9Y P9Y P9{?. Y P }9Y P9Y P9Y P9 JKW - e(y) ?-Y P9Rshfe hftbercthkfs kf gkta shnb w.r.t y
Y P9{?. l 0 lkfstcft kj hftbercthkfs } 9
Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
60/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb :=-KJ-1=
Y
P9
]dclhfe tab vcdub kj bq-E hf bq-B , sk ,
Y P9Y P9 Cfswbr , hts lkrrblt cfswbr ,---------------jhfhsabn abrb----------
-------------Bx # 3.2 jhfhsabn abrb -------------
Y 3.: P stcrt abrb
HF^BEWC^HFE-JCL^KW\ YH.J 0 J(x,y) P?-
Hj wb acvb tab uf-bxclt nhjjbrbfthcd bqucthkf-C cfn ht
hftbercthfe jcltkr J(x,y)9 (h-b)
Y P9 hts cf uf-bxclt-nhjjbrbfthcd-bqucthkfs .
Ehvbf Hftbercthfe-Jcltkr J(x,y) ?-^act hs , -6 Y J(x,y)0Hftbercthfe-Jcltkr 0 HJ P9
^abf, jkr `cihfe bq-C cs cf bxclt nhjjbrbfthcd bqucthkf, sk,
tabf wb sakudn `udthpdy bq-C gy hts Hftbercthfe Jcltkrs Y
J(x,y) P.
^abf, wb ebt cf bxclt nhjjbrbfthcd bqucthkf Yh-bP
Y P9htskur bxclt nhjjbrbfthcd bqucthkf.
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
61/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb :3-KJ-1=
^abf wb lcf skdvbn ht gy tab `btakn kj sblthkf # 3.2,
\k, J(x,y) hs lcddbn cs Hftbercthfe jcltkr kj bq-C,
-----------------------------
Bxc`pdb # 3 ?-
\akw tact jkddkwhfe nhjjbrbfthcd bqucthkf-C hs fkt bxclt, gut ht
acs cf Hftbercthfe jcltkrs J(x,y) cfn skdvb tab fbw bqucthkf ;
Y P9Y P9hts fkt cf bxclt nhjjbrbfthcdbq.
\AKU ^AC^ ^AB BSRC^HKF-C H\ FK^ CF BQCL^NHJJBWBF^HCD BSRC^HKF?
\k, Y P9Uabrb, Y@(x,y)0 y 0
P9Y F(x,y) 0 - x 0
P9Jkr`udc uf-Bxclt nhjjbrbfthcd bqucthkfs ?
Y P9Y
P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
62/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb :7-KJ-1=
Cdsk,
Y P9Gy pdclhfe tab cgkvb vcdub hf bqucthkf-F,
Y P9Y
P9 Ht sakw tact Bq-C hs fkt bxclt nhjjbrbfthcd
bqucthkfs .
FKU @CIHFE BS-C CF BQCL^-NHJJBWBF^HCD BSRC^HKFGX @RD^H]DXHFE BSRC^HKF-C UH^A H^\ HF^BEWC^HFE-
JCL^KW\YHJP J(x,y) ?-
Y
P9
\k, Y P9Gy `udthpdyhfe J(x,y) whta Bq-C , tabf Bq-C gblk`bs,
Y P9Y
P9
Y P9Y P9Y P9
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
63/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb :
-
7/29/2019 eng-math-3, ch-1, 1st order ODE ,
64/94
YC]]DHBN-BFEHFBBWHFE-@C^AB@C^HL\-HHH, LA-3 HF G@B, JHW\^-
KWNBW-KWNHFCWX-NHJJBWBF^HCD-BSRC^HKF\9 pcrt-3P
Y
Y G H K - @ B N H L C D - B F E H F B B W H F E - E R H N B . H F L G X \ H I C F N B W - D K N A H P ,w w w . ` b n h l c d - h ` c e b - p r k l b s s h f e . g d k e s p k t . l k ` ]ceb :>-KJ-1=
Bxc`pdb # 7 ?-Zbrhjy tact J(x)0 x