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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.

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When you ask this kind of question on MO, users automatically want to know where the equation comes from. It's only natural curiosity.... So, would you like to share? – Thierry Zell Apr 21 2011 at 1:50
Neither Maple nor Mathematica liked your equation, I'm not overly optimistic. – Thierry Zell Apr 21 2011 at 2:01
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 Closed form, not much chance. Analytic, yes. Existence theorem should be in all of the rigorous ODE textbooks. – Gerald Edgar Apr 21 2011 at 2:48
Gerald, lately I have been noticing questions where the word "analytic" is used as a synonym for "closed form," with the likelihood that the OP does not know basic existence and uniqueness facts. I think we are seeing that here. – Will Jagy Apr 21 2011 at 2:59
To Thierry: The question is from my current finance model research. I come up with this ODE by myself, not from anywhere else. To Gerald: Thank you for your answer. Do you have any suggestion what method should I use to solve it? Or give the name of an ODE book? I have tried to find some books in the library, but failed to find similar example. To Jary: I will try to improve my math skills. I am not a mathematician. I asked some of my math friends, but failed to get the solution. – William Apr 21 2011 at 3:35
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closed as too localized by Andres Caicedo, Will Jagy, S. Carnahan♦ Apr 21 2011 at 5:29

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