PDE $au_{xx} +2bu_{xy}+cu_{yy}$ into canonical form

Question

I am looking for the right substitution, to transform the pde $au_{xx} +2bu_{xy}+cu_{yy}=f$ with $ac \gt b^2$ into the canonical form $\nabla^2U(\hat x,\hat y)=F(\hat x,\hat y)$ where $\nabla^2$ is the laplace-operator. Using the linear substituion $\begin{pmatrix}\hat x \\ \hat y \end{pmatrix}=M\begin{pmatrix} x \\ y \end{pmatrix}$

Would appreciate any hint/help

Answer 1

After the change of variables

$$ r = m_1 x + m_2 y\\ s = m_3 x + m_4 y $$

we got

$$ (am_3^2+m_4(2bm_3+c m_4))u_{ss}+2(am_1 m_3+b(m_2 m_3+m_1m_4)+cm_2m_4)u_{rs}+(am_1^2+2bm_1m_2+c m_2^2)u_{rr}=0 $$

I hope this helps.