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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.

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