11.5.3.6 With variable coefficients

Problem: Determine the type of the differential equation, bring it to the canonical form, write down the general solution, find the solution of the Cauchy problem. \[ \begin{array}{l} y^{4} u_{x x}+2 y^{2} u_{x y}+u_{y y}-\frac{2}{y} u_{y}=0, \\ u(x, 1)=\frac{x^{3}}{3}, \quad u_{y}(x, 1)=2 x . \end{array} \]