Diffusion equn needed to be solved is $$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}[D(x)\frac{\partial p(x,t)}{\partial x}] \tag{1},$$
where $D(x)=a(1+bx)$ and given BC and IC are $p(x,t=0)=K\delta(x)$ and $j(x=x_1,t)=j(x=\infty,t)=0$ ,
where $j(x,t)$ is given by $$j(x,t)=-D(x)\frac{\partial p(x,t)}{\partial x} \tag{2}.$$
I tried this problem by separation of varibale by putting $p(x,t)=F(x)G(t)$ , I got the solution for $t$ part but for spatial part I got the ode: $$a(1+bx)\frac{d^2 F}{dx^2}+ab\frac{dF}{dx}+mF=0 \tag{3}$$
Now I can't find a closed form solution of equn(3) to which I can apply BC and IC ,though after making some change of varibale I reduced equn(3) to Bessel ODE but even then I can't apply conditions. How to proceed ... thanks in advance.