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I'm having some problems in finding the general solution of a linear second-order hyperbolic PDE in canonical form $$ u_{xy} = F(u_x, u_y, u, x, y) $$ where $F$ is some function.

Specifically I'm interested in solving this PDE in canonical form $$ u_{xy} + k(u_x + u_y) + (k^2-\sigma^2P(x-y))u = f(x)f(y) $$ where $k$, $\sigma$ are constant parameters and $P(x)$, $f(x)$ are some known function.

More than knowing the general solution of the second equation I'm mainly interested in knowing what is the general strategy for solving the first, generic, linear second-order hyperbolic PDE.

Most material I have read suggests only to find the canonical form of the equation (which I was able to do) but does not give further instructions on how to solve this canonical equation, but I have the impression there must be some cookbook recipe for doing so.

Thank you very much!

pp.ch.te
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    You can basically feed it through method of characteristics. – user10354138 Oct 27 '23 at 08:29
  • Thanks for the comment. Could you please elaborate on that? – pp.ch.te Oct 27 '23 at 08:31
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    Strictly speaking, general solution for second (and higher) order equations looses its importance since only in very few case it can be explicitly found (and the general hyperbolic equation is not one of them). What you really can search for is the solution of well defined problems, i.e. solution which satisfy additional conditions as the Cauchy problem, the Dirichlet and Neumann problem and so on. In your specific case, if you need a solution of the Cauchy problem or of the Goursat problem you can use Riemann's method to solve it. – Daniele Tampieri Nov 05 '23 at 13:14

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