y''+y'=x^2

y''+y'=0

characteristic equation

r^2+r=0

r=0，r=- 1

y=C 1e^(-x)+C2

Let y''+y' = x 2 have a special solution y = ax 3+bx 2+CX.

y'=3ax^2+2bx+c

y''=6ax+2b

6ax+2b+3ax^2+2bx+c=x^2

3a= 1，2b+6a=0 2b+c=0

a= 1/3，b=- 1，c=2

Special solution y = (1/3) x 3-x 2+2x

Y''+y' = x 2 general solution y = (1/3) x 2-x 2+2x+c1e (-x)+C2.

2

y''+2y'+y=cosx

y''+2y'+y=0

Characteristic equation r 2+2r+ 1 = 0.

r=- 1

y=C 1e^(-x)+Cxe^(-x)

Let y''+2y'+y=cosx special solution y=mcosx+nsinx.

y'= -msinx+ncosx

y''= -mcosx -nsinx

-MC OSX-nsinx-2 msinx+2 ncosx+MC OSX+nsinx = cosx

(-m+2n+m)= 1 (-n-2m+n)=0

m=0，n= 1/2

y=( 1/2)sinx

The general solution y = c1e (-x)+cxe (-x)+(1/2) sinx.

x=0，y=0 C 1=0

y'=-c 1e^(-x)+ce^(-x)+cxe^x+( 1/2)cosx x = 0，y ' =-c 1+c+( 1/2)= 3/2 c = 1

Special solution y = xe (-x)+( 1/2) sinx