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I have a Hamiltonian

$ -\sqrt{x^2 + y^2 - p^2} $

Hamilton-Jacobi equation for this becomes:

$ \frac{\partial\varphi}{\partial x} - \sqrt{x^2 + y^2 - (\frac{\partial\varphi}{\partial y})^2} = 0 $

I know that a candidate $\varphi(x,y)$ of the form $ c_1 x^2 + c_2 xy + c_3 y^2 $ can solve this thing, because I can do the partial derivation, plug it in, and find values for my constants that make it equal zero. But I do not have enough experience solving PDEs to solve for $\varphi$ in the most general way.

How can we solve a nonlinear, non-separable PDE like this given a candidate form that works?

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