How do I solve this second order PDE?
$\frac{\partial^2 X}{\partial t^2}+X= \cos(\omega t)+\frac{1}{4}(1-exp(-\tau +F(\sigma)))^\frac{-3}{2} [\sin(6t-6\psi)+3\sin(4t-4\psi)+3\sin(2t-2\psi)]$
$\omega$ is a constant. $F(\sigma)$ is just a general function of $\sigma$. $\tau$ and $\sigma$ are slow time scales, i.e. $\epsilon t$ and $\epsilon^2 t$ respectively where $\epsilon$<<1. And $\psi(\sigma)$ is just a function of $\sigma$.
Is it ok if I turn this into second order ordinary differential equation? And instead of constants, use general functions of $\sigma$ and $\tau$.
$\frac{d^2 X}{dt^2}+X= \cos(\omega t)+\frac{1}{4}(1-exp(-\tau +F(\sigma)))^\frac{-3}{2} [\sin(6t-6\psi(\sigma))+3\sin(4t-4\psi(\sigma))+3\sin(2t-2\psi(\sigma))]$
Then solving the homogeneous part gives me X = $F_1 (\tau,\sigma)\cos(t)+F_2 (\tau,\sigma)\sin(t)$.
What would be the particular solution for the non-homogeneous part?