11.4.38 First order partial differential equations

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ x y^{2} \frac{\partial u}{\partial x}+2 y^{3} \frac{\partial u}{\partial y}=2\left(y u-x^{2}\right), \quad y=1, \quad u=2 x^{2} . \]