11.4.31 First order partial differential equations

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