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I understand that the method of characteristics can be used to obtain analytical solutions of any first order linear P.D.E with constant or variable coefficients. But in all the examples and proofs that I've seen, the equations involved are all of degree $1$. And as far as I've tried, I can't seem to make this method work for equations of a higher degree.

Can the method of characteristics be tweaked in some way for solving equations like $a(u_x)^2 + b(u_y)^2 +cu =0$? If not, how does one solve such equations?

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  • The method of characteristics gets more involved, but is still workable. See F. John's book or http://en.wikipedia.org/wiki/Monge_cone or http://en.wikipedia.org/wiki/Method_of_characteristics#Fully_nonlinear_case – dls Sep 21 '13 at 04:00

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