11.5.3.4 With variable coefficients

Problem: Find the solution of the second order linear equation, satisfying the given initial conditions (the Cauchy problem). \[ \begin{array}{l} z_{x x}-2 x z_{x y}+x^{2} z_{y y}-z_{x}+(x-1) z_{y}=0, \\ z(0, y)=y, \quad z_{x}(0, y)=y^{2} . \end{array} \]