Analytic Solution of a Parabolic Partial Differential Equation with Up to Quadratic Spatial Coefficients What is the analytic solution for the following diffusion partial differential equation (initial value problem) of $f(t,x)$?
$$\frac{\partial f}{\partial t} = \frac12c_3^2x^2\frac{\partial^2 f}{\partial x^2}+(c_1-c_2x)\frac{\partial f}{\partial x},$$
where $c_i$'s are real number constants.
We can Fourier transform $f$ in the time $t$ domain and obtain the ODE of the transformed function $\tilde f(k,x)$ for any frequency $k$
$$ik\tilde f = \frac12c_3^2x^2\frac{\partial^2\tilde f}{\partial x^2}+(c_1-c_2x)\frac{\partial\tilde f}{\partial x}.$$
We can also apply the Laplace transform with respect to $t$ with parameter $k$ and the ODE of the transformed function $\hat f(k,x)$
$$k\hat f-f(0,x) = \frac12c_3^2x^2\frac{\partial^2\hat f}{\partial x^2}+(c_1-c_2x)\frac{\partial\hat f}{\partial x}.$$
How do we transform the first ODE into the standard form of hypergeometric equation? How do we deal with the second equation?
 A: Your Fourier transformed equation has solutions involving Kummer M and U functions:
$$
{x}^{-{\frac {-{c_{{3}}}^{2}+\sqrt {{c_{{3}}}^{4}+ \left( 8\,ik+4
\,c_{{2}} \right) {c_{{3}}}^{2}+4\,{c_{{2}}}^{2}}-2\,c_{{2}}}{{2 c_{{3}}
}^{2}}}}{{ M}\left({\frac {-{c_{{3}}}^{2}+\sqrt {{c_{{3}}}^{4}
+ \left( 8\,ik+4\,c_{{2}} \right) {c_{{3}}}^{2}+4\,{c_{{2}}}^{2}}-2\,c
_{{2}}}{2{c_{{3}}}^{2}}},\,{\frac {{c_{{3}}}^{2}+\sqrt {{c_{{3}}}^{4}+
 \left( 8\,ik+4\,c_{{2}} \right) {c_{{3}}}^{2}+4\,{c_{{2}}}^{2}}}{{c_{
{3}}}^{2}}},\,2\,{\frac {c_{{1}}}{{c_{{3}}}^{2}x}}\right)}
$$
and the same with $U$ instead of $M$.
EDIT:
This arises in the following way.  If you write $\tilde{f}(k,x) = x^r g(s/x)$, the equation becomes 
$$ g''(v) = - \dfrac{2 r c_1}{s c_3^2 v} g(v) + \dfrac{c_3^2 (r-r^2) + 2 i k + 2 r c_2}{c_3^2 v^2} g(v) + \left( 2 \dfrac{c_1}{s c_3^2} + 2 \dfrac{(r-1) c_3^2 - c_2}{c_3^2 v}\right) g'(v) $$
That is the Kummer differential equation (with the appropriate values of the parameters) if $r$ is chosen to make $c_3^2(r-r^2)+2ik+2rc_2 = 0$ and $s$ is chosen to make $2 c_1/(s c_3^2) = 1$.
