# Solving eigenvalue equation involving exponential

I would like to solve for the following eigenvalue equation: $$A\frac{d^2}{dx^2}f(x) + B(1-e^{-Cx})^2 f(x) = \mu f(x)$$ $A<0$, $B>0$, and $C>0$ are real, positive constants with $\mu$ being the eigenvalue. My text indicates that this equation can be solved exactly with eigenvalue of the form $$\mu_n = \alpha \left(n+\frac{1}{2}\right) + \beta \left(n+\frac{1}{2}\right)^2$$ where $\alpha$ and $\beta$ some constants. I would appreciate any help.