Number of zeros of a solution of an ODE

Question

Consider $f\in C^1(\mathbb{R},\mathbb{R})$ and $y$ a non-null solution of equation \begin{equation} y''=f(y(t)), \qquad t \in [0,1] \end{equation} Prove that $y$ has just a finite number of zeros, if any. Do you have any suggestion?

Answer 1

Hint: Assume the contrary and use compactness to find $t_n\rightarrow t^*$ for which $y(t_n)=0$ (whence also $y(t^*)=0$). Show that $y'(t^*)=0$ and also $y''(t^*)=0$ and that this implies $y\equiv 0$ contradicting the hypothesis.