Post Closed as "too localized" by Andres Caicedo, Will Jagy, S. Carnahan
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# Does this ODE question have analytic solution?

These days, I am struggling with following ODE problem when I build up my research model:

$1/2f''(x)+a(b - x) f'(x) -(c+ e^{A+Bx})f(x)=0$ where f(x) is a smooth function, and $a,b,c, A,B$ are all constants. How to get the closed form of f(x)?

I tried the Laplace transform to work on it, say $F(s) = L(f(x))$, but because of $e^{A+Bx}$, there will be a term $F(s-B)$ in the transformed equation. How to deal with this term?

I also tried the power series method, but got some very complicate coefficients, which stops me going further.

I think the term $e^{A+Bx}$ is the difficult part.

Could anyone here tell me how to deal with this kind of problem? Does the solution exit? I tried several ODE books but cannot find similar examples. Or could any one can suggest some relevant books?

Thank you very much.