Consider the Autonomous non linear System $$ x'= y \\ y'=-ky+h(x) $$ with $k \in \mathbb R, k\gt 0$ being a real parameter and $h:\mathbb R \to \mathbb R $ a continuous function of $x$ such that $ h(0)=0$ and $ h(x) \neq 0$ for every other $x \neq 0$, and also $h$ is not linear in $x$ (otherwise the whole system would be linear). I have to determine whether $(0,0)$ is a stable equilibrium point.
My attempt: I have tried different ways to solve this. If I study the Jacobian matrix, I need $h'(x) $ (at least $h'(0) $) but $h$ is not said to be differentiable. If I try to find a Lyapunov function, I don't know how to deal with $h$ in a neighborhood of $(0,0)$. I even tried to determine a function $g$ such that $(x', y') = \nabla g $ and a possible Hamiltonian of the system but it didn't work.
What would you do? Thank you.