a, b, c, d are arbitrary constants then the corresponding differential equation to $y=a{e}^x+b{e}^{-x}+c\cos x+d\sin x$  is 

$y=a{e}^x+b{e}^{-x}+c\cos x+d\sin x\to \left(1\right)$
Diff.
$\begin{array}{rl} & y^I=a{e}^x-b{e}^{-x}-c\sin x+d\cos x\\ & y^{II}=a{e}^x+b{e}^{-x}-c\cos x-d\sin x\\ & y^{III}=a{e}^x-b{e}^{-x}+c\sin x-d\cos x\\ & y^{IV}=a{e}^x+b{e}^{-x}+c\cos x+d\sin x\end{array}$
$y^{\left(4\right)}=\left(y\right)$