Question

Solve the given differential equation by undetermined coefficients. y'' + 6y' + 9y = −xe^6x

Answer 1

noter for diggerential equatiatus gcall y" + py't qy where p q are constad, The complete soutien e whene, He & the general solution of the bomegaseous equatico y" tpy't qy =0 and Up å lhe particular solutions of the bon-hamogeneous equation, y" + py't ay = grai).

To solve the given dipperential equation by the method of undetermined coepgo ciente gh toy't ay Giren, 62 ze Solution: consider the differential equation, y" toy' + ay = -ae 68

To find ye ,3 6 atb The auxiliary equalién is 10+ cm+950 Cro+cm+30 103-3,-3 (repeated roots) The general solution of the bomogeneous equation , Y = Ge +62 212321 To find up: we have, let guz - ale Y = Aneon + Bega p = CAR+B) eon La de?+ GALE"+ 63.63 = CA+GB+GADEGA 621 * = 6A269 + CARS" + 36AMES 6 36 Be 21 = ( 24 + 363 + 36A2de DOW cubatitullogtry'-y in we have 691 C12A +373 + 36A " + 6y + ay E-2 262 + 6CA+GB+6A2066 +90A2+B) ega -ae6n 62 12A +36 B + 36A2 + 6A + 368 + 36 A2 + 9A2 tabe -2e 1067 C188 + 81 B + B14D) 69 - aega 18 A + BIB + BLA = -21

To find ye ,3 6 atb The auxiliary equalién is 10+ cm+950 Cro+cm+30 103-3,-3 (repeated roots) The general solution of the bomogeneous equation , Y = Ge +62 212321 To find up: we have, let guz - ale Y = Aneon + Bega p = CAR+B) eon La de?+ GALE"+ 63.63 = CA+GB+GADEGA 621 * = 6A269 + CARS" + 36AMES 6 36 Be 21 = ( 24 + 363 + 36A2de DOW cubatitullogtry'-y in we have 691 C12A +373 + 36A " + 6y + ay E-2 262 + 6CA+GB+6A2066 +90A2+B) ega -ae6n 62 12A +36 B + 36A2 + 6A + 368 + 36 A2 + 9A2 tabe -2e 1067 C188 + 81 B + B14D) 69 - aega 18 A + BIB + BLA = -21

Companing the like terms, 18A + BLB = we have SIA - now, SLA = -1 AE- A 18 A + 813 = 0 28-4) +818.30 87. 9 B = 2 9 = 2 729 The particular dution in Up - tnega + 2 ega ST T29 Therefore, The general cotedion y = Yetup 8, -32 -32 tale y = Ge 691 Lale 81 '+ 2061 *729