11.5.3.5 With variable coefficients

Problem: Find the solution of the second order linar equation, satisfying the given initial conditions (the Cauchy problem). \[ \begin{array}{l} y^{2} z_{x x}+2 y \cdot z_{x y}+z_{y y}+z_{x}=0, \\ z(x ; 0)=x^{3}, \quad z_{y}(x ; 0)=-x . \end{array} \]