Using change of variables to change PDE to form with no second order derivatives

[lefturner]

Homework Statement

Classify the equation and use the change of variables to change the equation to the form with no mixed second order derivative. u_{xx}+6u_{xy}+5u{yy}-4u{x}+2u=0

Homework Equations

I know that it's of the hyperbollic form by equation a_{12}^2 - a_{11}*a_{22}, which gives 4 >0 so it's hyperbollic.

The Attempt at a Solution

What I don't understand is how to get started on changing the form to something with no second order derivatives. My book used a matrix method but shows no examples. Could I have a tip for where to get started? Thanks!

 

[HallsofIvy]

You don't want "no second order derivatives", you want "no second order mixed derivatives". In other words, you want to get rid of that "u_{xy}" term.

Think of this as x^2+ 6xy+ 5y^2, a quadratic polynomial.

Rotating through an angle \theta would give x= x'cos(\theta)- y'sin(\theta), y= x'sin(\theta)+ y'cos(\theta). Put those into the polynomial and choose \theta so that the x'y' term is 0.

Equivalently, simpler calculations but more sophisticated, write the polynomial as
\begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}1 & 3 \\ 3 & 5\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}
Use the eigenvectors of that matrix as the new axes.