I am a mathematical physics student and I had the following question in my mind from few weeks. I couldn't find any solutions. I am very thankful to this site and I hope, I expect some good reasonable solution.
I would like to know the solution of equation in the form of:
$$\frac{\partial^2 w}{\partial t \partial x} = F\left(w, \frac{\partial w}{\partial t},\frac{\partial w}{\partial x}, \frac{\partial^2 w}{\partial t^2},\frac{\partial^2 w}{\partial^2 x}\right)$$
Please generalize by an example.
Thanks in advance.
as per the expert request I am giving a problem in the above form:
Find general solution of $$ \left(\frac{\partial w}{\partial x}\right)^2 \frac{\partial^2 w}{\partial t^2} = \left(\frac{\partial w}{\partial t}\right)^2 \frac{\partial^2 w}{\partial x^2}$$ and my second problem is $$2\frac{\partial w}{\partial t} \frac{\partial w}{\partial x}\frac{\partial^2 w}{\partial t\partial x} = \left(\frac{\partial w}{\partial x}\right)^2\frac{\partial^2 w}{\partial t^2} + \left(\frac{\partial w}{\partial t}\right)^2\frac{\partial^2 w}{\partial x^2}$$
Thanks in advance