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I am trying to solve a second order parabolic PDE using the method of characteristics. The PDE has the following form:

$$\alpha\frac{\partial^2u}{\partial x^2}-\gamma\frac{\partial u}{\partial x}-\frac{\partial u}{\partial y}-f(u(x,y))=0.$$ where $\alpha$ and $\gamma$ are constants and $f$ is a non-linear function.

If I follow the standard procedure, I have to build $\Delta=b^2-ac=0$ which shows the PDE is parabolic. $$\Rightarrow \frac{dy}{dx}=\frac{b+\sqrt{\Delta}}{a}=0$$ Now it seems that no new variable can be chosen. Do you any idea how this problem can be solved...?

Thanks in advance

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