# Solve y′′+2y′+y={(4y^t,0<=t<1),(0,t>=1):} where y(0)=0,y′(0)=0

Solve ${y}^{″}+2{y}^{\prime }+y=\left\{\begin{array}{ll}4{e}^{t},& 0\le t<1\\ 0,& t\ge 1\end{array}$ where $y\left(0\right)=0,{y}^{\prime }\left(0\right)=0$
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anekesanxl
$\begin{array}{rl}\left(\mathcal{L}f\right)\left(s\right)=& {\int }_{0}^{\mathrm{\infty }}f\left(t\right){e}^{-st}\phantom{\rule{thinmathspace}{0ex}}dt\\ =& {\int }_{0}^{1}4{e}^{t}{e}^{-st}\phantom{\rule{thinmathspace}{0ex}}dt\\ & \text{and so on.}\end{array}$