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

## 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!

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
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.