11.2.2 Nonlinear equations

Problem: Determine the type of the nonlinear equation, based on the given solution: \[ \begin{array}{l} u_{x x}^{2}+u_{x y}^{2}+u_{y y}^{2}+u_{x}+u_{y}=0, \\ u=x^{2}-2 x y+y^{2}-6 x-6 y+5 . \end{array} \]