Differential Equations (MTH401) Lecture That a non-homogeneous linear differential equation of order n is an equation of the form n

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1 Diffrntial Equations (MTH40) Ltur 7 Mthod of Undtrmind Coffiints-Surosition Aroah Rall. That a non-homognous linar diffrntial quation of ordr n is an quation of th form n n d d d an + a a a0 g( ) n n = n d d d Th offiints a, a, 0, a n an b funtions of. Howvr, w will disuss quations with onstant offiints.. That to obtain th gnral solution of a non-homognous linar diffrntial quation w must find: Th omlmntar funtion, whih is gnral solution of th assoiatd homognous diffrntial quation. An artiular solution of th non-homognous diffrntial quation.. That th gnral solution of th non-homognous linar diffrntial quation is givn b Gnral solution = Comlmntar funtion + Partiular Intgral Finding Comlmntar funtion has bn disussd in th rvious ltur. In th nt thr lturs w will disuss mthods for finding a artiular intgral for th nonhomognous quation, naml Th mthod of undtrmind offiints-surosition aroah Th mthod undtrmind offiints-annihilator orator aroah. Th mthod of variation of aramtrs. Th Mthod of Undtrmind Coffiint Th mthod of undtrmind offiints dvlod hr is limitd to non-homognous linar diffrntial quations That hav onstant offiints, and Whr th funtion g() has a sifi form. Coright Virtual Univrsit of Pakistan 48

2 Diffrntial Equations (MTH40) Th form of g() Th inut funtion g() an hav on of th following forms: A onstant funtion k. A olnomial funtion An onntial funtion Th trigonomtri funtions sin( β ), os( β ) Finit sums and roduts of ths funtions. Othrwis, w annot al th mthod of undtrmind offiints. Th mthod Consist of rforming th following sts. St Dtrmin th form of th inut funtion g(). St Assum th gnral form of aording to th form of g() St Substitut in th givn non-homognous diffrntial quation. St 4 Simlif and quat offiints of lik trms from both sids. St 5 Solv th rsulting quations to find th unknown offiints. St 6 Substitut th alulatd valus of offiints in assumd Rstrition on g? Th inut funtion g is rstritd to hav on of th abov statd forms baus of th rason: Th drivativs of sums and roduts of olnomials, onntials t ar again sums and roduts of similar kind of funtions. Th rssion a + b + has to b idntiall qual to th inut funtion g(). Thrfor, to mak an duatd guss, is assurd to hav th sam form as g. Caution! In addition to th form of th inut funtion g(), th duatd guss for must tak into onsidration th funtions that mak u th omlmntar funtion. No funtion in th assumd must b a solution of th assoiatd homognous diffrntial quation. This mans that th assumd should not ontain trms that duliat trms in. Taking for grantd that no funtion in th assumd is duliatd b a funtion in, som forms of g and th orrsonding forms of ar givn in th following tabl. Coright Virtual Univrsit of Pakistan 49

3 Diffrntial Equations (MTH40) Numbr Th inut funtion g() Trial artiular solutions An onstant.g. A A + B A + B A + B + C + D 5 sin 4 A os 4 + B sin 4 6 os 4 A os 4 + B sin A 5 8 ( 9 ) 5 ( A + B) 5 Th assumd artiular solution 9 5 ( A + B + C) 5 0 sin 4 A os4 + B sin 4 sin 4 ( A + B+ C )os4 + ( A + B + C )sin4 5 os4 ( A + B) os 4 + ( C + D) sin 4 If g( ) Suos that quals a sum? ( ) Th inut funtion g onsists of a sum of m trms of th kind listd in th abov tabl i.. g( ) = g( ) + g ( ) + + gm ( ). Th trial forms orrsonding to g ( ), g ( ),, gm ( ) b,,,. Thn th artiular solution of th givn non-homognous diffrntial quation is = m In othr words, th form of is a linar ombination of all th linarl indndnt funtions gnratd b ratd diffrntiation of th inut funtion g(). m Coright Virtual Univrsit of Pakistan 50

4 Diffrntial Equations (MTH40) Eaml Solv + 4 = + 6 Comlmntar funtion To find, w first solv th assoiatd homognous quation + 4 W ut = m, = m m, = m m Thn th assoiatd homognous quation givs ( m + 4m ) m Thrfor, th auiliar quation is m + 4m as m 0, Using th quadrati formula, roots of th auiliar quation ar m = ± 6 Thus w hav ral and distint roots of th auiliar quation m = 6 and m = + 6 Hn th omlmntar funtion is ( + = 6) ( + + 6) Nt w find a artiular solution of th non-homognous diffrntial quation. Partiular Intgral Sin th inut funtion g( ) = + 6 is a quadrati olnomial. Thrfor, w assum that = A + B + C Thn Thrfor = A + B and = A + 4 = A + 8A + 4B A B Substituting in th givn quation, w hav A + 8A + 4B A B C = + 6 or A + (8A B) + (A + 4B C) = + 6 Equating th offiints of th lik owrs of, w hav C Coright Virtual Univrsit of Pakistan 5

5 Diffrntial Equations (MTH40) - A =, 8A - B = -, A + 4B - C = 6 Solving this sstm of quations lads to th valus A =, B = 5, C = 9. Thus a artiular solution of th givn quation is 5 = 9. Hn, th gnral solution of th givn non-homognous diffrntial quation is givn b = + or = ( + 6) + ( + 6) Eaml Solv th diffrntial quation + = sin Comlmntar funtion To find, w solv th assoiatd homognous diffrntial quation + Put = m, = m m, = m m Substitut in th givn diffrntial quation to obtain th auiliar quation ± i m m + or m = Hn, th auiliar quation has oml roots. Hn th omlmntar funtion is ( ) = + os sin Partiular Intgral Sin sussiv diffrntiation of g( ) = sin rodu sin and os Thrfor, w inlud both of ths trms in th assumd artiular solution, s tabl Coright Virtual Univrsit of Pakistan 5

6 Diffrntial Equations (MTH40) = Aos + Bsin. Thn = Asin + B os. = 9Aos 9Bsin. Thrfor + = ( 8A B)os + (A 8B)sin. Substituting in th givn diffrntial quation ( 8A B)os + (A 8B)sin os + sin. From th rsulting quations 8 A B, A 8B = Solving ths quations, w obtain A = 6 7, B = 6 7 A artiular solution of th quation is 6 6 = os sin 7 7 Hn th gnral solution of th givn non-homognous diffrntial quation is ( ) 6 6 = os sin + + os sin 7 7 Eaml Solv = Comlmntar funtion To find, w solv th assoiatd homognous quation Put = m, = m m, = m m Substitut in th givn diffrntial quation to obtain th auiliar quation m m ( m + )( m ) m =, Thrfor, th auiliar quation has ral distint root m =, m = Thus th omlmntar funtion is = +. Partiular intgral Sin g ( ) = (4 5) + 6 = g( ) + g ( ) Corrsonding to g ( ) = A + B Corrsonding to g ( ) = ( C + D) Coright Virtual Univrsit of Pakistan 5

7 Diffrntial Equations (MTH40) Th surosition rinil suggsts that w assum a artiular solution = + i.. A B C D = + + ( + ) Thn A C D C = + ( + ) + C D C = 4 ( + ) + 4 Substituting in th givn 4D Simlifing and grouing lik trms = 4C + 4D C + 4C A 4C A B C D = A A B C + (C D) = Substituting in th non-homognous diffrntial quation, w hav A A B C + (C D) = Now quating onstant trms and offiints of, and, w obtain A B = 5, A = 4 C = 6, C D Solving ths algbrai quations, w find A = 4, B = 9 C =, D = -4 Thus, a artiular solution of th non-homognous quation is = ( 4 ) + ( 9) (4 ) Th gnral solution of th quation is. = + = + ( 4 ) + ( 9) - (4 ) Duliation btwn and? If a funtion in th assumd is also rsnt in thn this funtion is a solution of th assoiatd homognous diffrntial quation. In this as th obvious assumtion for th form of is not orrt. In this as w suos that th inut funtion is mad u of trms of n kinds i.. g ) = g ( ) + g ( ) + + g ( ) ( n and orrsonding to this inut funtion th assumd artiular solution is = n If a ontain trms that duliat trms in, thn that must b multilid with i n, n bing th last ositiv intgr that liminats th duliation. i Coright Virtual Univrsit of Pakistan 54

8 Diffrntial Equations (MTH40) Eaml 4 Find a artiular solution of th following non-homognous diffrntial quation To find 5 = + 4 8, w solv th assoiatd homognous diffrntial quation W ut = m in th givn quation, so that th auiliar quation is Thus Sin Thrfor, m 5m = + g( ) = 8 = A m =, 4 Substituting in th givn non-homognous diffrntial quation, w obtain A 5 A + 4A = 8 So 0 = 8 Clarl w hav mad a wrong assumtion for, as w did not rmov th duliation. Sin A is rsnt in. Thrfor, it is a solution of th assoiatd homognous diffrntial quation To avoid this w find a artiular solution of th form = A W noti that thr is no duliation btwn and this nw assumtion for Now = A + A, = A + A Substituting in th givn diffrntial quation, w obtain A + A 5A 5A + 4A = 8. or A = 8 A = 8. So that a artiular solution of th givn quation is givn b = (8 ) Hn, th gnral solution of th givn quation is = + 4 (8 ) Coright Virtual Univrsit of Pakistan 55

9 Diffrntial Equations (MTH40) Eaml 5 Dtrmin th form of th artiular solution (a) = 5 7 (b) + 4 = os. (a) To find w solv th assoiatd homognous diffrntial quation Put = m Thn th auiliar quation is m 8m + 5 m = 4 ± i Roots of th auiliar quation ar oml 4 = ( os + sin ) Th inut funtion is g( ) = 5 7 = (5 7) Thrfor, w assum a artiular solution of th form A B C D = ( ) Noti that thr is no duliation btwn th trms in and th trms in. Thrfor, whil roding furthr w an asil alulat th valu A, B, C and D. (b) Considr th assoiatd homognous diffrntial quation + 4 Sin g( ) = os Thrfor, w assum a artiular solution of th form = ( A + B)os + ( C + D) sin Again obsrv that thr is no duliation of trms btwn and Coright Virtual Univrsit of Pakistan 56

10 Diffrntial Equations (MTH40) Eaml 6 Dtrmin th form of a artiular solution of + = 5sin+ 7 6 To find, w solv th assoiatd homognous diffrntial quation + Put = m Thn th auiliar quation is ± m m + m = i Thrfor = ( ) os + sin 6 Sin g ( ) = 5sin+ 7 = g( ) + g( ) + g( ) Corrsonding to g ( ) = : = A + B + C Corrsonding to g ( ) = 5sin : = D os + E sin Corrsonding to g = : = ( F + G) 6 ( ) 7 6 Hn, th assumtion for th artiular solution is = + + or = A + B + C + D os + E sin + ( F + G) No trm in this assumtion duliat an trm in th omlmntar funtion 7 = + Eaml 7 Find a artiular solution of + = Considr th assoiatd homognous quation + m Put = Thn th auiliar quation is m m + = ( m ) m =, Roots of th auiliar quation ar ral and qual. Thrfor, = + 6 Coright Virtual Univrsit of Pakistan 57

11 Diffrntial Equations (MTH40) Sin g ( ) = Thrfor, w assum that = A This assumtion fails baus of duliation btwn and. W multil with Thrfor, w now assum = A Howvr, th duliation is still thr. Thrfor, w again multil with and assum = A Sin thr is no duliation, this is atabl form of th trial = Eaml 8 Solv th initial valu roblm + = 4 + 0sin, ( π ), ( π ) = Solution Considr th assoiatd homognous diffrntial quation + m Put = Thn th auiliar quation is m + m = ± i Th roots of th auiliar quation ar oml. Thrfor, th omlmntar funtion is = os + sin Sin g ( ) = 4 + 0sin = g( ) + g( ) Thrfor, w assum that = A + B, = C os + Dsin So that = A + B + C os + Dsin Clarl, thr is duliation of th funtions os and sin. To rmov this duliation w multil with. Thrfor, w assum that So that = A + B + C os + Dsin. = Csin Cos + Dos Dsin + = A + B C sin + D os Substituting into th givn non-homognous diffrntial quation, w hav A + B C sin + D os = 4 + 0sin Equating onstant trms and offiints of, sin, os, w obtain Coright Virtual Univrsit of Pakistan 58

12 Diffrntial Equations (MTH40) B, A = 4, C = 0, D So that A = 4, B, C = 5, D Thus = 4 5 os Hn th gnral solution of th diffrntial quation is = + = os + sin + 4-5os W now al th initial onditions to find and. ( π ) osπ + sinπ + 4π 5π osπ Sin sin π,osπ = Thrfor = 9π Now = 9π sin + os sin 5os Thrfor ( π ) = 9π sinπ + osπ π sinπ 5osπ = = 7. Hn th solution of th initial valu roblm is = 9π os + 7sin os. Eaml 9 Solv th diffrntial quation = 6 + Th assoiatd homognous diffrntial quation is = m Put = Thn th auiliar quation is m 6m + 9 m = Thus th omlmntar funtion is = + 0, Sin g( ) = ( + ) = g( ) + g ( ) W assum that + Corrsonding to g ( ) = : Corrsonding to g ( ) = : Thus th assumd form of th artiular solution is = A + B + C + D = A + B + C D Th funtion in is duliatd btwn and. Multiliation with dos not rmov this duliation. Howvr, if w multil with, this duliation is rmovd. Thus th orativ from of a artiular solution is = A + B + C + D = Coright Virtual Univrsit of Pakistan 59

13 Diffrntial Equations (MTH40) Thn and = = A + B + D + D A + D + 6D + 9D Substituting into th givn diffrntial quation and ollting lik trm, w obtain 6 + = 9A + ( A + 9B) + A 6B + 9C + D = 6 + Equating onstant trms and offiints of, and ilds A 6B + 9C =, A + 9B 9A = 6, D = Solving ths quations, w hav th valus of th unknown offiints A =, B = 8 9, C = and D = -6 Thus 8 = Hn th gnral solution 8 = + = Highr Ordr Equation Th mthod of undtrmind offiints an also b usd for highr ordr quations of th form n n d d d an + a... a a0 g( ) n n = n d d d with onstant offiints. Th onl rquirmnt is that g() onsists of th ror kinds of funtions as disussd arlir. Eaml 0 Solv + = os To find th omlmntar funtion w solv th assoiatd homognous diffrntial quation + Put = Thn th auiliar quation is m + m m m, = m, = m or m ( m + ) m,0, Th auiliar quation has qual and distint ral roots. Thrfor, th omlmntar funtion is = + + Sin g( ) = os Thrfor, w assum that = A os + B sin Clarl, thr is no duliation of trms btwn and. m Coright Virtual Univrsit of Pakistan 60

14 Diffrntial Equations (MTH40) Substituting th drivativs of trms, w hav in th givn diffrntial quation and grouing th lik + = ( A + 4B) os + ( 4A B) sin = Equating th offiints, of os and sin, ilds A + 4B =, 4A B Solving ths quations, w obtain A = 0, B = 5 So that a artiular solution is (0) os + = + + ( 5) Hn th gnral solution of th givn diffrntial quation is = + + (0) os + Eaml Dtrmin th form of a artiular solution of th quation + = Considr th assoiatd homognous diffrntial quation + Th auiliar quation is 4 m + m m, 0, 0, Thrfor, th omlmntar funtion is = ( Sin g ) = = g ( ) + g ( ) Corrsonding to g ( ) = : Corrsonding to g ) = : ( = A = B Thrfor, th normal assumtion for th artiular solution is = A + B Clarl thr is duliation of (i) Th onstant funtion btwn and. (ii) Th onntial funtion btwn and. sin ( 5) os. sin To rmov this duliation, w multil with and with. This duliation an t b rmovd b multiling with and artiular solution is = A + B. Hn, th orrt assumtion for th Coright Virtual Univrsit of Pakistan 6

15 Diffrntial Equations (MTH40) Eris Solv th following diffrntial quations using th undtrmind offiints = = = = os = ( )sin 6. 5 = = (os sin ) Solv th following initial valu roblms = ( + ), ( 0 ) =, ( 0 ) = 5 9. d + ω = F0 osγt, (0), (0) dt = 5 + 8, ( 0 ) = 5, ( 0 ) =, (0) = 4 Coright Virtual Univrsit of Pakistan 6