I am struggling with a second order PDE problem of three variables. I have been given the equation
$$ u_{xx} -3u_{xy}+u_{yy}-4u_{xz}+u_{zz}+u_{y}+4u_{z}=0$$
and asked to classify this as a elliptic, parabolic or hyperbolic.
Secondly, if the equation is not elliptic than a change of variables should be found to make it elliptic.
My take:
I am not sure about how to classify this as it is in three variables, but I think I should collect all second order coefficients in a three by three matrix and later check if it is positive definite. The matrix is
$$ A = \left( \begin{matrix} 1 & -3 & -4 \\ -3 & 1 & 0 \\ -4 & 0 & 1 \end{matrix} \right) $$
which is indefinite. Is this correct so far? Assuming it is, a change of variables is needed. I do not know how to make such a change of variables, anyone who can explain this?
Note: This is a homework problem, I would appreciate explanations rather than straight answers.