eng-math-3, ch-1, 1st order ODE ,

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


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


3/94



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>

@BÂKN-^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 ÂB 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

@BÂKN JKW JHFNHFE HF^BEWC^HFE-JCL^KW\ YHJP ?-............................................................. :5


4/94



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


5/94



Y


\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


6/94



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.


7/94



Y


----------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


35/94



Y



36/94



Y



37/94



Y



38/94



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

------------------------------


39/94



Y



40/94



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


41/94



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


42/94



Y


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


43/94



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 >


44/94



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?-


45/94



Y


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 ?-


46/94



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 ?-


47/94



Y


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


48/94



Y


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,


49/94



Y


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


50/94



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


51/94



Y


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?-


52/94



Y


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,


53/94



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


54/94



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


55/94



Y


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


56/94



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


57/94



Y


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 .


58/94



Y


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


59/94



Y


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


60/94



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) ?-âct hs , -6 Y J(x,y)0Hftbercthfe-Jcltkr 0 HJ P9

âbf, jkr `cihfe bq-C cs cf bxclt nhjjbrbfthcd bqucthkf, sk,

tabf wb sakudn ùdthpdy bq-C gy hts Hftbercthfe Jcltkrs Y

J(x,y) P.

âbf, wb ebt cf bxclt nhjjbrbfthcd bqucthkf Yh-bP

Y P9htskur bxclt nhjjbrbfthcd bqucthkf.


61/94



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=

âbf 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 ÂC^ ÂB 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

P9Jkrùdc uf-Bxclt nhjjbrbfthcd bqucthkfs ?

Y P9Y

P9


62/94



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,

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Y P9Y

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bqucthkfs .

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JCL^KW\YHJP J(x,y) ?-

Y

P9

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Y P9Y

P9

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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 :


64/94



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

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eng-math-3, ch-1, 1st order ODE ,