Question: Find the solution of the following Cauchy problems:

(c) \(x u_{x}+y u_{y}=2 x y\), with \(u=2\) on \(y=x^{2}\),

(e) \(y u_{x}+x u_{y}=x y, x \geq 0, y \geq 0\), with \(u(0, y)=\exp \left(-y^{2}\right)\) for \(y>0\), and \(u(x, 0)=\exp \left(-x^{2}\right)\) for \(x>0\)

(j) \(\sqrt{x} u_{x}+u u_{y}+u^{2}=0, \quad u(x, 0)=1,0

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