Second-order linear

For a second-order linear homogeneous equation $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$, try solutions of the form $y=e^{rx}$. Substitution gives the characteristic equation $a{r}^2+br+c=0$. If its roots $r_1,r_2$ are real and distinct, the general solution is $y=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$. If there is a repeated root $r$, the general solution is $y=(C_1+C_{2}x)e^{rx}$. If the roots are complex $r=\alpha \pm \beta i$, the general solution is $y=e^{\alpha x}(C_1\cos (\beta x)+C_2\sin (\beta x))$. Use the initial conditions to solve for $C_1$ and $C_2$.