11.4.1 First order partial differential equations

Problem: a) Find the general solution of the first-order partial differential equation: \[ x \frac{\partial u}{\partial x}-u \frac{\partial u}{\partial y}=0, \quad(x>0) \] b) Solve the first-order partial differential equation under the given additional conditions: \[ x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=u-x^{2}-y^{2} ; y=-2, \quad u=x-x^{2} . \]