Question

Find the appropriate form of a particular solution $y_{p}(x)$ using method of undetermined coefficients:

$y''+4y'+4y=2xe^{-2x}+8sin(2x)$

(do not solve for coefficient constants)

Answer 1

let Solicy" + 4y'+4y= 2x1-23 +2 31(20) The characteristic equ is 1+49+4=0 = - 4116-411) (4) => A=- 2 - 2 Ye (X) = 4e-2x + Cg xe 2x yola)= (A +BX+Cx2+D%3De-2x + E Smil2x) + F C03123) 4p [%) = (B+ 2C%+ 3D X2)e-2X – 2 (A+Bee+cm+D23)e-27 +2E cos (2x) - 2f sin (20) Yp" () = (22+6 Dx) -2X – 21B+ 2CX+ 3D*%)e-27 +4e-2% (A+Bx+ca?+Dx3) -21B+ 204+ 3D*%) e-27 - 2 E sin (2x) - 4F cos (2x) se put them in to get A = B = C = 0, D = }, E=0, F = -1 to get Yp(x)= e-2x x cos(2x) es particular cold 3 3