solving a piecewise ODE

Question

In mechanics the simple equation $\ddot{x}+2\zeta\omega_n\dot{x}+\omega_n^2x=0$ is often used to model a mass-spring-damper system where no external force is used and the solution to this equation is very easy to calculate and well known, then ICs can just be inputted and its sorted. However, I was wondering what I can do in this situation if I were to model the damper more realistically, that is the damper constant is not the same both directions of motion: $$c=\begin{cases}c_1&\dot{x}>0\\0&\dot{x}=0\\c_2&\dot{x}<0\end{cases}$$ this is defined in the equation as: $$\zeta=\frac{c}{2m\omega_n},\omega_n=\sqrt{\frac km}$$ any sorces on this kind of modelling or approaching this problem would be appreciated, Thanks!