second order ODE with variable coefficients 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. 

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 $C_i$.
