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I have a partial differential equation as follows: $$\frac{\partial p(x,t)}{\partial t}=\text{Dp} \frac{\partial ^2p(x,t)}{\partial x^2}-\frac{p(x,t)-\text{p0}}{\tau }$$ What I try to do was to get its general solution by

DSolve[D[p[x, t], t] == -((p[x, t] - p0)/τ) + Dp D[p[x, t], {x, 2}], p, {x, t}]

However, the output returns what I have inputted, which means Mathematica cannot solve this PDE.

By looking up the help document, I find that DSolve can only find the general solution for a restricted type of homogeneous linear second-order PDEs, that is, for such a linear second-order PDE as $b \frac{\partial ^2u}{\partial x\, \partial y}+a \frac{\partial ^2u}{\partial x^2}+c \frac{\partial ^2u}{\partial y^2}+d \frac{\partial u}{\partial x}+e \frac{\partial u}{\partial y}+f u=g$, only when d=0,e=0,f=0 and a,b and c are all constants can we get the general solution.

I have tried this PED in Maple, and get a solution with conditions, but I don't know whether it is correct. The help document tells that the algorithm used by DSolve is not applicable in this case.

So, I wonder whether there exist some other ways or some packages to solve this PDE. If finding the general solution is impossible, is it possible to get a special solution?

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You can refer to this answer. – xzczd Mar 11 '14 at 2:48
None of us could make a living doing PDEs if it was that easy.. – Kai Sikorski Mar 11 '14 at 6:15
and get a solution with conditions, but I don't know whether it is correct. You can always substitute the solution back in the PDE and see if it satisfies it? Or since you used Maple, you can use pdetest – Nasser Mar 11 '14 at 6:33
@xzczd Thanks for this information. I'll try it again. – Z-Y.L Mar 11 '14 at 6:59
@Nasser Thank you! – Z-Y.L Mar 11 '14 at 7:00

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