Question

Solve the given differential equation by undetermined coefficients.
y'' − 2y' + 2y = e^2x(cos(x) − 8 sin(x))

Answer 1

y"-2y! +2y = c2* (1052 - 8 Sima) Total solution gex) = get up 10 find yo, D2-20+2=0 then rie, m²- 2m +2=0. ma at 14-8 marbtJ6²400 m= 2+54 2+2 malt If m= x+B then yc = exx [Crosßi + (2. Sinky] 14 yeret fcorresten] Yea ex{co cosa + Casinx] of HP - 621 ( COD - SS0%). 02-2D+2 up = I en Ven) then yp= e eax 1 Va) fcpta), (cosa assinx) (0+2 2 2 (0+2)+2 - - LO, 2) 12. (cosa 8 sina) Total 2x YP = 14 +2 D240+4-20 get) = - (cost-8 Sira) D2 +20+2

1 COSX sim] you can be cosa] - 8e" [ 8e2x [ Sina] ID²+20+2 LD?+20+2 COS(91+b) (a) Sincaut) fio put D²= -02 then 1 .-1+2D12 bati 20-1 Sim -1+20+2 1 (204) I 2D + (2D-u - 8eruf en (2041) (20-1 7P 2. SinB] D'--12 -1 you can play cosu 1 - Pen[ baby som] L -4-1 - D=. we know that DC Cosa) = -Sim (si) = cosa you e Casim -6054] 5[ acosa-sine} y=-22 [-2.sina-c05x. = +6C05x + Esimm] you meza (-17605* + 6 Sina? you ea[ 140054- Esim]. Total solution, ya) - Ye typ yeu) = ex [c1c054 Casinu] + eran [ 14.6052-6 sina]