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Quasilinear Equation

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I have to solve the following quasilinear equation using the method of characteristics:

xu_x + yu_y = xy
u|y=x^2 = sinx
dx/dt = x gives me lnx = t +c_1
dy/dt= y gives me lny = t + c_2

But what do I do with du/dt = xy - yu?

Thank you.

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The expert solves quasilinear equations using the methods of characteristics.

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