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I am trying to solve the following 2nd order PDE analytically, but haven't succeeded so far. I have tried to separate the variables, but it doesn't work here. I would be very grateful for any suggestions that could lead me in the right direction in solving it and for all suggestions about where I could look further. The equation is the following:
$$\frac{\partial u(x,t)}{\partial x}=- \frac{b(x)}{2} (1-u(x,t))^2+\frac{b(x)}{c} \frac{ \partial^2 u(x,t)} {\partial x \partial t},$$ where $u(x,t)$ is the unknown function, $b(x)$ is a known function of $x$ only, and $c$ is a constant.

Any help would be much appreciated. Thanks!

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