Consider the following non-linear PDE:
$u_x^2+u_y^2-u_{xx}-u_{yy}=2a+bz$
where $x,y,z$ are independent variables with $u=u(x,y)$ and $a,b$ are constants.
I am trying to find at least one examples of $u(x,y)$ which satisfies the above PDE. Can anyone help me to solve the above PDE? I have also tried to solve the same by Mathematica and Matlab. But I could not succeed. Also, is there any suggestions/references for solving any non-linear PDE?
Well, you can try some easy things if you ignore some variables. For example: $$u(x,y,z)=x\sqrt{2a+bz}$$ may work, at least for $2a+bz>0$.
Furthermore, you can try to solve $v_x^2-v_{xx}=1$ (This can be solved in Wolfram: https://www.wolframalpha.com/input/?i=%28du%2Fdx%29%5E2-d%5E2u%2Fdx%5E2%3D1) and put: $$ u=(2a+bz)\cdot v $$ which will work too.