Ordinary Differential Equations/Non Homogenous 4

Solve
1. $$y''-y=e^{-x},x\in\mathbb R, y(0)=y'(0)=0$$

Answer: $$y(x)=\frac14 e^x -\frac14 e^{-x} -\frac12 xe^{-x}$$

Find Particular Integral
1. $$y''-7y+12y=24e^x$$

Answer: $$y_p(x)=4e^{-x}$$

2. $$ y''+4y'+4y=e^{-2x}$$

Answer: $$y_p(x)=\frac12 x^2e^{-2x}$$