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.