11.5.4.1 With constant coefficients

Problem: Bring the equation to canonical form and find its general solution: \[ a_{11} u_{x x}^{\prime \prime}+2 a_{12} u_{x y}^{\prime \prime}+a_{22} u_{y y}^{\prime \prime}+a_{10} u_{x}^{\prime}+a_{20} u_{y}^{\prime}=0 \] \begin{tabular}{|c|c|c|c|c|} \hline\( a_{11} \) & \( a_{12} \) & \( a_{22} \) & \( a_{10} \) & \( a_{20} \) \\ \hline 1 & 4 & 12 & \( 1 / 6 \) & 1 \\ \hline \end{tabular}