\begin{equation}
2 y^{\prime \prime}+y^{\prime}-4 y=0, y(0)=0, y^{\prime}(0)=1
\end{equation}
\begin{equation}
\begin{array}{l}{y=e^{r t}} \\ {y^{\prime}=r e^{n}} \\ {y^{\prime \prime}=r^{2} e^{r t}}\end{array}
\end{equation}
\begin{equation}
\begin{array}{l}{e^{n} \neq 0} \\ {r(r+3)=0}\end{array}
\end{equation}
\begin{equation}
r=0,-3
\end{equation}
\begin{equation}
y=c_{1}+c_{2} e^{-3 t}
\end{equation}
\begin{equation}
y(0)=-2
\end{equation}
\begin{equation}
-2=c_{1}+c_{2}
\end{equation}
\begin{equation}
y^{\prime}=-3 c_{2} e^{-3 t}
\end{equation}
\begin{equation}
\begin{array}{l}{y^{\prime}(0)=3} \\ {3=-3 c_{2}} \\ {\text { i.e. } c_{2}=-1}\end{array}
\end{equation}
\begin{equation}
\begin{array}{l}{\text { i.e. } c_{2}=-1} {c_{1}=-1}\end{array}
\end{equation}
\begin{equation}
y=-1-e^{-3 t}
\end{equation}
