Finding a boundary of a two-point boundary problem knowing one boundary and a solution

Question

Suppose that we have a function $y(t)$ solution of the following ODE: $$ a(t).Y''(t) + b(t).Y'(t) + c(t).Y(t) = F(t)$$ Suppose also that

1. there exist two reals $z_0$ and $z_1$ such that $y(z_0)=y'(z_0)=0$ and $y(z_1)=y'(z_1)=0$
2. $y(t)$ has no other real zeros than $z_0$ and $z_1$
3. $z_0$ is known but not $z_1$

The question is: is it possible to find (or numerically approximate) $z_1$ by using the known information, i.e. $y(t)$, $z_0$, $F(t)$, $a(t)$, $b(t)$, $c(t)$ ?

Standard root finding algorithms can't be really used here since there is no reasonable initial guess for $z_1$ and $y(t)$ could be not smooth as well in general.

I wonder if the Sturm-Liouville theory could help here since it looks like a Two-Point Boundary Values problem but where we try to find a boundary knowing the other boundary and the solution.