Solve $y'' - y' - 2y = cosh2t$
Firstly, consider homogeneous equation $y'' - y' - 2y = 0$, the characteristic polynomial is
\begin{align*}
r^2 - r - 2 &= 0\\
(r + 1)(r - 2) &= 0
\end{align*}
Therefore, the solution to homogeneous equation is $c_1 e^{-t} + c_2 e^{2t}$
Then need to solve $y'' - y' - 2y = cosh2t$, since
\begin{equation*}
cosh(2t) = \frac{e^{2t} + e^{-2t}}{2}
\end{equation*}
we come to guess $y = Ate^{2t} + Be^{-2t}$ is a solution.
\begin{align*}
y' &= Ae^{2t} + 2Ate^2t - 2Be^{-2t}\\
y'' &= 2Ae^{2t} + 2Ae^{2t} + 4Ate^{2t} + 4Be^{-2t}\\
&= (4A + 4At)e^{2t} + 4Be^{-2t}
\end{align*}
then plug $y$, $y'$ and $y''$ back in $y'' - y' - 2y$, we get
\begin{align*}
y'' - y' - 2y =& (4A + 4At)e^{2t} + 4Be^{-2t} - (Ae^{2t} + 2Ate^2t - 2Be^{-2t}) - 2(Ate^{2t} + Be^{-2t}) \\
&= (4A - A)e^{2t} + (4A - 2A - 2A)te^{2t} + (4B + 2B - 2B)e^{-2t}\\\
&= 3Ae^{2t} + 4Be^{-2t}
\end{align*}
since $y'' - y' - 2y = cosh2t$
\begin{align*}
&3Ae^{2t} + 4Be^{-2t} = \frac{e^{2t} + e^{-2t}}{2}\\
&\implies 3A = \frac{1}{2}\quad and\quad 4B = \frac{1}{2}\\
&\implies A = \frac{1}{6}\quad and \quad B = \frac{1}{8}
\end{align*}
Therefore the solution to $y'' - y' - 2y = cosh2t$ is $y = \frac{1}{6}te^{2t} + \frac{1}{8}e^{-2t}$
Then, the general solution is
\begin{equation*}
y = c_1 e^{-t} + c_2 e^{2t} + \frac{1}{6}te^{2t} + \frac{1}{8}e^{-2t}
\end{equation*}
