"Consider a first order linear PDE. (e.g. y u(adsbygoogle = window.adsbygoogle || []).push({}); _{x}+ x u_{y}= 0)

If u(x,y) is constant along the curves y^{2}- x^{2}= c, then this implies that the general solution to the PDE is u(x,y) = f(y^{2}- x^{2}) where f is an arbitrary differentiable funciton of one variable. We call the curves along which u(x,y) is constant thecharacteristic curves."

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I don't understand the implication above highlighted in red. The idea of characteristic curves seems to be very important in solving first order linear PDEs, but I am never able to completely understand the idea of it. Why would finding the characteristic curves help us find the general solution to the PDE?

Thanks for explaining!

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# First order linear PDE-the idea of characteristic curves

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