Implicit definition of a coordinate in differential equation

Question

I would like to solve the following differential equation

$\frac{d^2 \psi(r_*)}{dr^2_*} + (\omega^2 - V(r)) \psi(r_*)=0$

and the boundary conditions are $\psi(\inf) = \psi(-inf) = 0$. The problem is that the function V(r) is defined in $r$, not $r_*$. However I have the following relation

$r_* = r+2M ln(r-2M)$

which, clearly, cannot be solved for $r$.

My real problem is to find the eigenvalues for $\omega$, but any help how to deal with this implicit definition of $r$ may help me.

PS: I am posting this here and not in physics/math exchange because the function $V(r)$ is complicated and I need to solve it computationally. And no, I do not want to transform the whole equation to the $r$ coordinate and solve it in there.