Ordinary Differential Equations

Transcription

1 Ordiary Diffrtial Equatio Aftr radig thi chaptr, you hould b abl to:. dfi a ordiary diffrtial quatio,. diffrtiat btw a ordiary ad partial diffrtial quatio, ad. Solv liar ordiary diffrtial quatio with fid cotat by uig claical olutio ad aplac traform tchiqu. Itroductio A quatio that coit of drivativ i calld a diffrtial quatio. Diffrtial quatio hav applicatio i all ara of cic ad girig. Mathmatical formulatio of mot of th phyical ad girig problm lad to diffrtial quatio. So, it i importat for gir ad citit to kow how to t up diffrtial quatio ad olv thm. Diffrtial quatio ar of two typ A) ordiary diffrtial quatio ODE) A ordiary diffrtial quatio i that i which all th drivativ ar with rpct to a igl idpdt variabl. Eampl of ordiary diffrtial quatio iclud d y y 0, 0), y0) 4, d d d d y d y d y 5 y i, 0), 0), y 0) 4 d d d d d Ordiary diffrtial quatio ar claifid i trm of ordr ad dgr. Ordr of a ordiary diffrtial quatio i th am a th hight drivativ ad th dgr of a ordiary diffrtial quatio i th powr of hight drivativ. Thu th diffrtial quatio, d y d y y d d d i of ordr ad dgr, whra th diffrtial quatio i i of ordr ad dgr. d d A gir approach to diffrtial quatio i diffrt from a mathmaticia. Whil, th lattr i itrtd i th mathmatical olutio, a gir hould b abl to itrprt th rult phyically. So, a gir approach ca b dividd ito thr pha: a) formulatio of a diffrtial quatio from a giv phyical ituatio, b) olvig th diffrtial quatio ad valuatig th cotat, uig giv coditio, ad c) Itrprtig th rult phyically for implmtatio

2 08.0. Chaptr 08.0 Formulatio of diffrtial quatio A dicud abov, th formulatio of a diffrtial quatio i bad o a giv phyical ituatio. Thi ca b illutratd by a prig-ma-dampr ytm. K b M Figur Sprig-ma dampr ytm. Abov i th chmatic diagram of a prig-ma-dampr ytm. A block i updd frly uig a prig. A mot phyical ytm ivolv om kid of dampig - vicou dampig, dry dampig, magtic dampig, tc., a dampr or dahpot i attachd to accout for vicou dampig. t th ma of th block b M, th prig cotat b K, ad th dampr cofficit b b. If w maur diplacmt from th tatic quilibrium poitio w d ot coidr gravitatioal forc a it i balacd by tio i th prig at quilibrium. Blow i th fr bo diagram of th block at tatic ad amic quilibrium. So, th quatio of motio i giv by Ma F S F D ) whr F S i th rtorig forc du to prig. F D i th dampig forc du to th dampr. a i th acclratio. Th rtorig forc i th prig i giv by F S K ) a th rtorig forc i proportioal to diplacmt ad it i gativ a it oppo th motio. Th dampig forc i th dampr i giv by F D bv ) a th dampig forc i dirctly proportioal to vlocity ad alo oppo motio. Thrfor, th quatio of motio ca b writt a Ma K bv 4)

3 Primr for Ordiary Diffrtial Equatio Static Dyamic T F S F D Mg Ma Figur Fr bo diagram of prig-ma-dampr ytm. Sic d d a ad v dt dt from Equatio 4), w gt d d M K b dt dt d d M b K 0 5) dt dt Thi i a ordiary diffrtial quatio of cod ordr ad of dgr o. Solutio to liar ordiary diffrtial quatio I thi ctio w dicu two tchiqu ud to olv ordiary diffrtial quatio A) Claical tchiqu B) aplac traform tchiqu Claical Tchiqu Th gral form of a liar ordiary diffrtial quatio with cotat cofficit i giv by d y d y d y k... k k k y F ) 6) d d d d Th gral olutio cotai two part

4 Chaptr 08.0 y y H y P 7) whr y H i th homogou part of th olutio ad y P i th particular part of th olutio. Th homogou part of th olutio y H i that part of th olutio that giv zro wh ubtitutd i th lft had id of th quatio. So, y H i olutio of th quatio d y d y d y k... k k k y 0 d d d d 8) Th abov quatio ca b ymbolically writt a D y kd y... kdy k y 0 9) D kd... kd k) y 0 0) whr, d D d ) d D d... opratig o y i th am a D r), D r ), D r ) opratig o aftr th othr i ay ordr, whr D r ), D r ),..., D r ) ar factor of D kd kd k 0 ) To illutrat D D ) y 0 i am a D ) D ) y 0 D ) D ) y 0 Thrfor, D kd... kd k) y 0 ) i am a D r ) D r )... D r ) y 0 4) opratig o aftr th othr i ay ordr.

5 Primr for Ordiary Diffrtial Equatio Ca : Root ar ral ad ditict Th tir lft had id bcom zro if D r y 0. Thrfor, th olutio to D r y 0 i a olutio to a homogou quatio. D r y 0 i calld ibitz liar diffrtial quatio of firt ordr ad it olutio i D r y 0 5) r y d 6) r d y 7) Itgratig both id w gt l y r c 8) r y c 9) Sic ay of th factor ca b placd bfor y, thr ar diffrt olutio corrpodig to diffrt factor giv by r r r r C, C,..., C, C whr r, r,....., r, r ar th root of Equatio ) ad C, C,..., C, C ar cotat. W gt th gral olutio for a homogou quatio by uprimpoig th idividual ibitz olutio. Thrfor r r r r yh C C C C 0) Ca : Root ar ral ad idtical If two root of a homogou quatio ar qual, ay r r, th D r ) D r )... D r ) D r ) y 0 ) t work at D r ) D r ) y 0 ) If D r ) y z ) th D r ) z 0 r z C 4) Now ubtitutig th olutio from Equatio 4) i Equatio ) r D r ) y C r r y C d r r r C y d

6 Chaptr 08.0 r d y) C d d r y) Cd 5) Itgratig both id of Equatio 5), w gt r y C C r y C C ) 6) Thrfor th fial homogou olutio i giv by r r r yh C C C... C 7) Similarly, if m root ar qual th olutio i giv by m rm rm r yh C C C... Cm Cm... C 8) Ca : Root ar compl If o pair of root i compl, ay r i ad r i, whr i th i i r r yh C C C... C 9) Sic i co ii, ad 0a) i co ii 0b) th r r yh C co ii C co ii C... C r r C C co i C C i C... C r r Aco Bi C... C ) whr A C C ad B i C C) ) Now, lt u look at how th particular part of th olutio i foud. Coidr th gral form of th ordiary diffrtial quatio D k D k D... k y ) X Th particular part of th olutio y P i that part of olutio that giv X wh ubtitutd for y i th abov quatio, that i, D k D k D k y X... 4) Sampl Ca a a Wh X, th particular part of th olutio i of th form A. W ca fid A by a ubtitutig y A i th lft had id of th diffrtial quatio ad quatig cofficit. P

7 Primr for Ordiary Diffrtial Equatio Eampl Solv d Solutio y, y 0) 5 Th homogou olutio for th abov quatio i giv by D y 0 Th charactritic quatio for th abov quatio i giv by r 0 Th olutio to th quatio i r y H C Th particular part of th olutio i of th form A da A d A A A A Hc th particular part of th olutio i y P Th complt olutio i giv by y y H y P C Th cotat C ca b obtaid by uig th iitial coditio y 0) C 5 y C 5 C 6 Th complt olutio i y 6 Eampl Solv. y 5, y 0) 5 d Solutio Th homogou olutio for th abov quatio i giv by D y 0 Th charactritic quatio for th abov quatio i giv by r 0 Th olutio to th quatio i r.5

8 Chaptr y H C Bad o th forcig fuctio of th ordiary diffrtial quatio, th particular part of th olutio i of th form A. 5, but ic that i part of th form of th homogou part of th olutio, w d to choo th t idpdt olutio, that i,.5 y P A To fid A, w ubtitut thi olutio i th ordiary diffrtial quatio a.5 A.5. d 5 A d A A A.5.5 A A 0.5 Hc th particular part of th olutio i.5 y P 0.5 Th complt olutio i giv by y y H y P.5.5 C 0.5 Th cotat C i obtaid by uig th iitial coditio y 0) 5..50).50) 0 C 0.50) 5 y C 0 5 C 5 Th complt olutio i.5 y Sampl Ca Wh X ia) or coa ), th particular part of th olutio i of th form Ai a) Bco a). W ca gt A ad B by ubtitutig y Ai a) Bco a) i th lft had id of th diffrtial quatio ad quatig cofficit. Eampl Solv d y.5y i, y 0) 5, 0) d d d Solutio Th homogou quatio i giv by D D.5) y 0 Th charactritic quatio i r r.5 0 Th root of th charactritic quatio ar

9 Primr for Ordiary Diffrtial Equatio r i i Thrfor th homogou part of th olutio i giv by 0.75 yh K co K i ) Th particular part of th olutio i of th form Ai Bco y P d d d d d Aco Bi Aco Bi ).5 Ai Bco ) i d Ai Bco ) Aco Bi ).5 Ai Bco ) i.5a B)i.5B A)co i Equatig cofficit of i ad co o both id, w gt.5a B.5B A 0 Solvig th abov two imultaou liar quatio w gt A B 0.97 Hc y P i 0.97co Th complt olutio i giv by 0.75 y K co K i ) i 0.97co ) To fid K ad K w u th iitial coditio y 0) 5, 0) d From y 0) 5 w gt Ai B co Ai B co.5 Ai B co ) i 0.750) 5 K co0) K i0)) i0) 0.97co0)) 5 K 0.97 K K co K i ) K i K co ) d co 0.97i From

10 Chaptr ), d w gt 0.750) 0.750) 0.75 K co0) K i0)) K i0) K co0)) co0) 0.97i0) 0.75K K ) K K Th complt olutio i 0.75 y 5.97co i ) i 0.97co Eampl 4 Solv d y 6.5y co ), y 0) 5, 0) d d d Solutio Th homogou part of th quatio i giv by D 6D.5) y 0 Th charactritic quatio i giv by r 6r r 6 ) 4).5) ,.956 Thrfor, th homogou olutio y H i giv by yh K K Th particular part of th olutio i of th form y P Ai Bco Subtitutig th particular part of th olutio i th diffrtial quatio, d d Ai Bco ) 6 Ai Bco ) d d.5 Ai Bco ) co d Aco Bi ) 6 Aco Bi ) d.5 Ai Bco ) co

11 Primr for Ordiary Diffrtial Equatio Ai Bco ) 6 Aco Bi ).5 Ai Bco ) co.5a 6B)i.5B 6A)co co Equatig cofficit of co ad i w gt.5b 6A.5A 6B 0 Th olutio to th abov two imultaou liar quatio ar A B Hc th particular part of th olutio i y P i co Thrfor th complt olutio i y y H y P y K K ) i co Cotat K ad K ca b dtrmid uig iitial coditio. From y 0) 5, y 0) K K K K Now ).956) K.956K d co i From 0) d K.956K K.956K K.956K W hav two liar quatio with two ukow K K K.956K Solvig th abov two imultaou liar quatio, w gt K K.744 Th complt olutio i y ) i co. Sampl Ca Wh X a i b or a co b, th particular part of th olutio i of th form

12 08.0. Chaptr 08.0 a Ai b Bco b), w ca gt A ad B by ubtitutig a y Ai b Bco b) i th lft had id of diffrtial quatio ad quatig cofficit. Eampl 5 Solv d y y 5.5 i, y 0) 5, 0) d d d Solutio Th homogou quatio i giv by D 5D.5) y 0 Th charactritic quatio i giv by r 5r r 5 ) 4).5) ,.5 Sic root ar rpatd, th homogou olutio y H i giv by.5) yh K K) Th particular part of th olutio i of th form yp Ai Bco ) Subtitutig th particular part of th olutio i th ordiary diffrtial quatio d d { Ai B co )} 5 { Ai B co )} d d.5{ Ai B co )} i d { Ai B co ) Aco B i )} d 5{ Ai B co ) Aco B i )}.5 Ai B co ) { Ai B co ) 5{ Ai B co ) Aco B i ) Aco B i ) Aco B i )} Ai B co ) Aco B i ) i.875 Ai B co ) Aco B i ) i.875a B) i A.875B) co i Equatig cofficit of co ad i o both id w gt A.875B 0.875A B Ai B co ) Ai B co )} i i

13 Primr for Ordiary Diffrtial Equatio Solvig th abov two imultaou liar quatio w gt A ad B 0.45 Hc, yp 0.454i 0.45co ) Thrfor complt olutio i giv by y y H y P.5 y K K ) 0.454i 0.45co Cotat K ad K ca b dtrmid uig iitial coditio, From y 0) 5, w gt Now K K 5.45 d.5k.5.5k.5 K co 0.45i ) From 0), w gt d.50).50).50).5k.5k 0) K.5 ) 0.454i 0.45co ) co0) 0.45i0)) 0.454i0) 0.45co0).5K K K K ) K.977 K Subtitutig K 5.45 ad K i th olutio, w gt.5 y ) 0.454i 0.45co ) Th form of th particular part of th olutio for diffrt right had id of ordiary diffrtial quatio ar giv blow a X 0 a a a b y P 0 b b a A ib ) Ai b) Bco b) a ib) a Ai b) Bco b)

14 Chaptr 08.0 aplac Traform cob ) Ai b) Bco b) a cob) a Ai b) Bco b) If y f ) i dfid at all poitiv valu of, th aplac traform dotd by Y ) i giv by 0 Y ) { f )} f ) d 5) whr i a paramtr, which ca b a ral or compl umbr. W ca gt back f ) by takig th ivr aplac traform of Y ). { Y )} f ) 6) aplac traform ar vry uful i olvig diffrtial quatio. Thy giv th olutio dirctly without th city of valuatig arbitrary cotat paratly. Th followig ar aplac traform of om lmtary fuctio )! ), whr 0,,,... a ) a a i a) a co a) a a ih a) a coh a) 7) a Th followig ar th ivr aplac traform of om commo fuctio a a, whr,,...! a a!

15 Primr for Ordiary Diffrtial Equatio i a a a co a a ih a a a coh at a b a a i b b a b a a co b i a a a 8) Proprti of aplac traform iar proprty If a, b, c ar cotat ad f ), g ), ad h ) ar fuctio of th [ af ) bg ) ch )] a f )) b g )) c h )) 9) Shiftig proprty If { f )} Y ) 40) th { f )} Y a) 4) Uig hiftig proprty w gt a! a, 0 b a i b a b a a co b a b b a ih b a b a a coh b a b 4) Scalig proprty If { f )} Y ) 4)

16 Chaptr 08.0 th { f a)} Y 44) a a aplac traform of drivativ If th firt drivativ of f ) ar cotiuou th { f )} f ) d 0 Uig itgratio by part w gt 0 f f ) ) f ) ) d ) f )... ) ) ) 0 f ) d ) f ) 0 45) f 0) f 0) f 0)... f 0) f ) d Y ) f 0) f 0) f 0)... f 0) 46) aplac traform tchiqu to olv ordiary diffrtial quatio Th followig ar tp to olv ordiary diffrtial quatio uig th aplac traform mthod A) Tak th aplac traform of both id of ordiary diffrtial quatio. B) Epr Y ) a a fuctio of. C) Tak th ivr aplac traform o both id to gt th olutio. t u olv Eampl through 4 uig th aplac traform mthod. Eampl 6 Solv d Solutio y, y 0) 5 Takig th aplac traform of both id, w gt y d [ Y ) y0)] Y ) Uig th iitial coditio, y 0) 5 w gt [ Y ) 5] Y ) 0

17 Primr for Ordiary Diffrtial Equatio ) Y ) ) Y ) 5 6 Y ) ) ) Writig th prio for Y ) i trm of partial fractio 5 6 A B ) ) 5 6 A A B B ) ) ) ) 5 6 A A B B 0 Equatig cofficit of ad giv A B 5 A B 6 Th olutio to th abov two imultaou liar quatio i A B 8 8 Y ) Takig th ivr aplac traform o both id 6 { Y )} Sic at a Th olutio i giv by y ) 6 Eampl 7 Solv. y 5, y 0) 5 d Solutio Takig th aplac traform of both id, w gt.5 y d [ Y ) y0)] Y ).5

18 Chaptr 08.0 Uig th iitial coditio y 0) 5, w gt [ Y ) 5] Y ).5 ) Y ) ) Y ) Y ).5) ) 0 6.5).5) 0 6.5) 5 8.5) Writig th prio for Y ) i trm of partial fractio 5 8 A B.5).5.5) 5 8 A.5A B.5).5) 5 8 A. 5A B 0 Equatig cofficit of ad giv A 5.5A B 8 Th olutio to th abov two imultaou liar quatio i A 5 B Y ).5.5) Takig th ivr aplac traform o both id { Y )}.5.5) Sic a ad a a ) Th olutio i giv by.5.5 y ) a

19 Primr for Ordiary Diffrtial Equatio Eampl 8 Solv d y.5y i, y 0) 5, 0) d d d Solutio Takig th aplac traform of both id d y.5y i d d ad kowig d y Y y0 0 d d Y y0 d i ) w gt Y ) y0) 0) Y ) y0).5y ) d Y ) 5 Y ) 5.5Y ).5Y ) 0.5Y ) Y ) ) 0 Y ) 0.5 Writig th prio for Y ) i trm of partial fractio A B C D A A B 0 B C C C D D.5D

20 Chaptr 08.0 A C B C D A.5C D B.5D Equatig trm of,, 0 ad giv A C 0 B C D A.5C D 0 B.5D Th olutio to th abov four imultaou liar quatio i A B C 0.97 D Hc Y ).5.5 { ) } { 0.75) } ) Y ) { 0.75) } ) { 0.75) } { 0.75) } ) ) Takig th ivr aplac traform of both id ) { Y )} { 0.75) } { 0.75) { Y )} { 0.75) } { 0.75) Sic a b a a co b b b a a i b i a a

21 Primr for Ordiary Diffrtial Equatio co a a Th complt olutio i Eampl 9 y ) co co i co i i 0.97co i Solv d y 6.5y co, y 0) 5, 0) d d d Solutio Takig th aplac traform of both id d y 6.5y co d d ad kowig d y Y y0 0 d d Y y0 d co ) w gt Y ) y0) 0) 6Y ) y0).5y ) d Y ) 5 6Y ) 5.5Y ) 6).5Y ) Y ) Y ) 6.5 Writig th prio for Y ) i trm of partial fractio A B C D

22 08.0. Chaptr 08.0 A A B 0 B C 6 6C C D 6D.5D A C B 6C D A.5C 6D B.5D Equatig trm of,, 0 ad giv A C 0 B 6C D 6 A.5C 6D B.5D 6 Th olutio to th abov four imultaou liar quatio i A B C D Th Y ) {.5) } {.5) ) Y ) {.5) } ) 0.97 {.5) } {.5) } } Takig th ivr aplac traform o both id ) 0.97 { Y )} {.5) } {.5) ) ) ) ) ) Sic

23 Primr for Ordiary Diffrtial Equatio a b a a coh b b a a ih b b i a a a co a a Th complt olutio i y ) coh0.8956) ih0.8956) co i co i Eampl i Solv d y y 5.5 i, y 0) 5, 0) d d d Solutio Takig th aplac traform of both id d y 5.5y i d d kowig d y Y y0 0 d d Y y0 d i ) ) w gt co

24 Chaptr 08.0 Y ) y0) 0) 5Y ) y0).5y ) d ) Y ) 5 5Y ) 5.5Y ) ) 5.5Y ) 0 ) ) 5).5Y ) Y ) Y ) 5.5 Writig th prio for Y ) i trm of partial fractio C A B C D 5.5 5C.5C D D.5D A A A B B B C A 5C D A B.5C 5D A B.5D B Equatig trm of,, 0 ad giv four imultaou liar quatio C A 0 5C D A B 5.5C 5D A B 8.5D B 6 Th olutio to th abov four imultaou liar quatio i A B C 0.45 D Th Y ) {.5.565)}.5)

25 Primr for Ordiary Diffrtial Equatio ) ) ) ) ) Y ) ) ) ) ).5) 5.45 Takig th ivr aplac traform o both id ) ) ) ) ) 5.45 )} { Y ) ) ) ) ) 5.45 Sic b b a a a co b b a b a i a a )! ) a a Th complt olutio i y i co ).5.5 ) 0.455i 0.45co

26 Chaptr 08.0 ORDINARY DIFFERENTIA EQUATIONS Topic A Primr o ordiary diffrtial quatio Summary Ttbook ot of a primr o olutio of ordiary diffrtial quatio Major All major of girig Author Autar Kaw, Prav Chalaai Dat Novmbr 0, 05 Wb Sit