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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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    Because of the $e^{A+Bx}$ coefficient, I would be surprised if there was a closed form solution. An approximate solution generated by power series would at least give you an idea about the general behavior of the solution. If you have some particular application in mind, perhaps more information about the parameters (their magnitudes, e.g.) can be used to simplify further?2011-04-21
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    I tried the power series method. But it is too complicate for my model... Thank you all the same.2011-04-22

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