y''-2y'+2y=x+1, y(0) = 3, y'(0)=0

I haven't done this in some time, please help.

I know I need to find the homogenous solution and add it to a particular solution.

Homogenous:

$\displaystyle r^2-2r+2 = 0 $

So $\displaystyle r = \{ 1+i , 1-i \} $, then the homogenous solution is $\displaystyle y_{h} = c_{1}e^{x}cosx+c_{2}e^xsinx $

Particular:

Let y = Ax + B, y'=A, y''=0

0 -2 (A) + 2(Ax+B) = x+1

A=1/2, B=1, $\displaystyle y_{p}= \frac {1}{2} x + 1 $

yh + yp =$\displaystyle c_{1}e^{x}cosx+c_{2}e^xsinx + \frac {1}{2} x + 1 $