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Theorem (Super Position):

If a second order homogeneous D.E. has 2 solutions $a,b$, then $a+b$ is also a solution.

To solve a second order linear constant coefficient homogeneous differential equation $a D_x^2 y + b D_x y + cy=0$ let $y=e^{rx}$. Then by plugging in we get:

\begin{indisplay} \begin{aligned} & a(r^2e^{rx}) + b(re^{rx}) + ce^{rx} = 0 \\ ... ... br + c) = 0 \\ & \Leftrightarrow ar^2 + br + c=0 \end{aligned}\end{indisplay}

which can be solved for $r=r_1,r_2$ $\Rightarrow$ $y=Ae^{rx} + Be^{rx}$ $\forall A,B$ by super position. If there is a repeated root ($r_1=r_2$), than let $y=Ae^{rx}+Bxe^{rx}$, plug in, and solve.