Question

6. Solve the differential equation by using the method of undermined coefficients: y" - 2y + y = 2e - 3e-

Answer 1

y" - Qy' ty: 2 ex- 3 ex. dry - 2 dy + y 2 0 2 - 3 e? dx dx I let d =D D?y - 2 Dyt y = 284-3 e-? [ (02-20 + 1) y = 26² - 3 exo y = C.F. + P. L is the solno calculation of .C.fi - cf => D² - go + 1 - 0 > m?.Zmo (m-1)2 = 0 cf = (a + x G) et

let y = AX + Bere y' = Aex - Be-x Y = AX + B ex putting all these in maino Differential eqh - 0 A ex+ Be-x) - 2 (Aen-Bex) + Apk + Bex=281-36 4Be-t . per-3e- I comparing UB - B CORVADIL- B = -3 Once Daily

Here with this quess we are not able to find arunce o we need to guess another o Particular integral with the help of cof. -4- Alex Benda y' = A ( xx ex + ex ) + B PP(-1)] y'= Axel & Apx - Bero y = A (a extex) + Aext Beu yll= Axex + 2 Ae X + Ber o Axex + 2 A QX + B e-x - 2 (Axe? ner + Axex & Bex -Be-x) - 20 - 30 x QAX ex tu bet 200 3 RL A - CORVADILLA A e B are not found

Another guess - yo - A ? ex Be-x у": A (12. e1t exy 27 ) - Ве - 1 0 1 = A (?ext exx 2 x) (ex, I-е) +ве - 7 ут: Ах? е? + Aa e1 + e 7 4 2Ах е? не - x у" : Ast? et + ЧАх е* + 2Ae” + Вес (Az ? ex + yax e1 toAet fBe - x) 0 - {( Az? et + 29 x et ) - Be- t Az? е? 4 Ве - 1 - 2 е? - Зех _ Эн? е? ч Hyet + 2A e 7+ Ве-т П - 2 в 1 - уни ет + 28 е-х + Ве: 1 . 2ex. Зе- 1 2 не? , чве QR2 eX зе on companing на кожа

2А 2 А - ч . . 3D В - 3 Уе - ? е - 3 е-х ч 9. с. бр. у (1x C ) е 1 xex 3