91Ó°ÊÓ

Solve \(u_{x}^{2}+y u_{y}-u=0\) with initial condition \(u(x, 1)=\frac{x^{2}}{4}+1\).

Solve the given initial value problem and determine the values of \(x\) and \(y\) for which it exists: (a) \(x u_{x}+u_{y}=y, \quad u(x, 0)=x^{2}\) (b) \(u_{x}-2 u_{y}=u, \quad u(0, y)=y\) (c) \(y^{-1} u_{x}+u_{y}=u^{2}, \quad u(x, 1)=x^{2}\)