# Change of variables in a partial differential equation

How to convert the following partial differential equation (pde) $$\frac{\partial V}{\partial t}=aV-as\frac{\partial V}{\partial s}-b^2s^2\frac{\partial^2 V}{\partial s^2}$$ into a pde of the form $$\frac{\partial u}{\partial t}=x^2\frac{\partial^2 u}{\partial x^2}+cx\frac{\partial u}{\partial x}$$ by some change of variables? Here $a,b,c$ are constants. (A hint should be enough.)

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@doraemonpaul, I think that's enough editing for a while. You are flooding the front page. –  Gerry Myerson Aug 21 '12 at 1:56

If you send $t\mapsto -t$, it will fix the sign.
An exponential integrating factor factor of $e^{At}$ will get rid of the $aV$ term.
Then you just need to send $s\mapsto\lambda x$ and compute the correct scaling factors.
Thanks. I think the last step should be rescaling the time: $t_1=b^2t$. –  TCL Nov 11 '10 at 18:28