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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

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This question is unanswerable as is. There is no way to give the solution to an unspecified equation. What equation are you interested in? –  Ron Maimon Nov 12 '11 at 23:10
Now I added the problems. Plz answer... –  dr.jgk Nov 13 '11 at 16:36
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migrated from physics.stackexchange.com Nov 13 '11 at 22:34

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1 Answer

I don't know if it's possible to give more general solutions, but for your first equation, there are solutions of the following forms: $$ w(x,t) = F(a x + b t)$$ and $$ w(x,t) = F((x + a)(t + b)) $$ where $a,b$ are arbitrary constants and $F$ is an arbitrary $C^2$ function. For your second equations, there are solutions of the following forms: $$ w(x,t) = F(a x + b t)$$ and $$ w(x,t) = F((x+a)/(t + b))$$

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