Question

求二阶常系数非齐次微分方程的通解,上图

Answer 1

令y' = v,y'' = v'

y'' - 1/x · y' = xe^x

v' - v/x = xe^x,e^∫ - 1/x dx = e^- ln|x| = 1/x

v'/x - v/x² = e^x

(v/x)' = e^x

v/x = ∫ e^x dx = e^x + C

y' = xe^x + Cx

y = ∫ xe^x dx + C∫ x dx = xe^x - ∫ e^x dx + Cx²/2 = xe^x - e^x + Cx²/2 + D

y = (x - 1)e^x + Ex² + D

C,D,E均为任意常数,E = C/2