Quien solución de calculo diferencial

Veamos...

$$\begin{align}&\mathcal{L}_s\{y''+y'+2y\}=\mathcal{L}_s\{x\}\\&\\&\mathcal{L}_s\{y''\}+\mathcal{L}_s\{y'\}+2\mathcal{L}_s\{y\}=\dfrac{1}{s^2}\\&\\&[s^2\mathcal{L}_s\{y\}-s y(0)-y'(0)]+[s\mathcal{L}_s\{y\}-y(0)]+2\mathcal{L}_s\{y\}=\dfrac{1}{s^2}\\&\\&[s^2\mathcal{L}_s\{y\}-2]+s\mathcal{L}_s\{y\}+2\mathcal{L}_s\{y\}=\dfrac{1}{s^2}\\&\\&(s^2+s+2)\mathcal{L}_s\{y\}=\dfrac{1}{s^2}+2\\&\\&\mathcal{L}_s\{y\}=\dfrac{1+2s^2}{s^2(s^2+s+2)}\\&\\&\mathcal{L}_s\{y\}=\dfrac{-1/4}{s}+\dfrac{1/2}{s^2}+\dfrac{1/4(s+7)}{s^2+s+2}\\&\\&y=-\dfrac{1}{4}+\dfrac{x^2}{2}+\dfrac{1}{4}e^{-x/2}\cos \dfrac{\sqrt7}{2}x+\dfrac{13}{4\sqrt7}e^{-x/2}\sin \dfrac{\sqrt7}{2}x\end{align}$$