11.5.4.20 With constant coefficients

Problem: Find the particular solution of the second order partial differential equation satisfying the given conditions: \[ \begin{array}{l} z_{x x}+2 z_{x y}-3 z_{y y}+z_{\mathrm{x}}-z_{y}=0, \\ z(0, y)=y, \quad z_{\mathrm{x}}(0, y)=0 . \end{array} \]