11.2.1 Nonlinear equations

Problem: Determine the type of the nonlinear equation based on the given solution: \[ u_{x x}+u_{x x} u_{x y}+u_{y y}^{2}+u_{x}+u_{y}=1 \text {, } \] a) \( u_{1}=x^{2}-2 x y+y^{2}+2 x-3 y+1 \), b) \( u_{2}=3 x-2 y \).