I have the following second order ODEs $$\begin{cases} \ddot{x}(t) = p(y(t), \dot{y}(t)) \, x(t)\\ \ddot{y}(t) = k_1 \dot{y}(t) + k_2 y(t) \end{cases}$$ with $k_1, k_2 > 0$, $p: \mathbb{R}^2 \rightarrow \mathbb{R}^+$ and the dot is the derivative with respect to $t$.
I need to prove that the evolution of $y(t)$, for $t \geq 0$, is a function of $x(t)$ (and of its derivative) under some conditions. I have the intuition that it is possible, e.g. if $x(t)$ is a 1-to-1 function of $t$ there exists the inverse $t = \xi(x)$ and thus $y(t) = y(\xi(x)) = y(x)$. I think requiring that $x(t)$ is monotonic is too much restrictive. Could you please help me?