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do you have any idea how I can solve this, this is part of my model and I would be happy to have a closed form solution for this

$A\{(1+b\gamma)g+\frac{(1+b\gamma)b\gamma}{2}{V}-r_f\}+A_V k(\alpha-V)+\frac{1}{2}A_{VV} \sigma^2 V=a^{\lambda(\gamma-1)}-a^\gamma$ where A is a function of V (changing variable) and all the others are constant.

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  • $\begingroup$ Expect a very complicated expression involving hypergeometric functions... $\endgroup$
    – mickep
    Commented Apr 2, 2015 at 5:51

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Let me rewrite the equation using simpler notations: $$ C_1 x y'' + (C_2 - C_3 x) y' + (C_4 + C_5 x) y = C_6, $$ where $C_i$ are the corresponding constants.

However, even in a very particular homogeneous case with $C_6 = 0$ and $C_1,\dots,C_5 = 1$, i.e. $$ x y'' + (1 - x) y' + (1 + x) y = 0, $$ WolframAlpha gets very "hard" solution. So, it looks unrealistic to obtain a general solution in a closed form for arbitrary coefficients $C_i$.

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