Ordinary Differential Equations/Non Homogenous 4

Solve

1. $y^{″}-y=e^{-x},x\in ℝ,y(0)=y^{\prime }(0)=0$

Answer: $y(x)=\frac{1}{4}{e}^x-\frac{1}{4}{e}^{-x}-\frac{1}{2}x{e}^{-x}$

Find Particular Integral

1. $y^{″}-7y+12y=24e^x$

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

2. $y^{″}+4y^{\prime }+4y=e^{-2x}$

Answer: $y_p(x)=\frac{1}{2}{x}^2{e}^{-2x}$