微分方程的通解公式,大一高数 求下列微分方程通解

y”+y’=2x^2.e^x

The aux. equation

p^2 +p=0

p(p+1)=0

p=0 or -1

yg = Ae^(-x) +B

yp= (Cx^2 +Dx+F)e^x

yp’=(Cx^2 +Dx+F + 2Cx+D)e^x =[Cx^2 +(2C+D)x +D+F]e^x

yp”=[Cx^2 +(2C+D)x +D+F + 2Cx + 2C+D]e^x =[Cx^2 +(4C+D)x + 2C+2D+F].e^x

yp”+yp’=2x^2.e^x

[Cx^2 +(4C+D)x + 2C+2D+F].e^x +[Cx^2 +(2C+D)x +D+F]e^x =2x^2.e^x

[ 2Cx^2 +(6C+2D)x + 2C+3D+2F] e^x =2x^2.e^x

=>

2C=2 and 6C+2D=0 and 2C+3D +2F=0

=>

C=1 and D=-3 and F=7/2

yp = (x^2 -3x+ 7/2).e^x

y”+y’=2x^2.e^x 的通解

y=yg+yp =Ae^(-x) +B +(x^2 -3x+7/2).e^x