\begin{equation}
\begin{array}{l}{y^{\prime \prime}+2 y^{\prime}+2 y=0} \\ {\text { The characteristic equation is, } r^{2}+2 r+2=0} \\ {r=\frac{-2 \pm \sqrt{4-8}}{2}} \\ {r=-1 \pm i}\end{array}
\end{equation}
\begin{equation}
\begin{array}{c}{\text { If the roots of charateristic equations are complex numbers } \lambda \pm i \mu, \mu \neq 0} \\ {\text { then the general equation is } y=c_{1}e^{\lambda t} \cos (\mu t)+c_{2}e^{\lambda t} \sin (\mu t)} \\ {\text { Compare } r=-1 \pm i \text { with } \lambda \pm i \mu} \\ {\text { we have } \lambda=-1 \quad \mu=1} \\ {\text { Subsitute the value of } \lambda=-1 \quad \mu=1} \\ {y=c_{1} e^{-t} \cos (t)+c_{2} e^{-t} \sin (t)}\end{array}
\end{equation}
