11.6.1 Systems of equations in partial derivatives of the first order

Problem: Determine the type of the system of linear partial differential equations of the first order: \[ \left\{\begin{array}{l} u_{x}+v_{x}-u_{y}+v_{y}=x y \\ 2 v_{y}-v_{x}+u_{x}+3 u_{y}=u \end{array}\right. \]