Hyperbolic DE

**dwsmith** · January 12th 2011, 01:25 PM

I don't understand the second step in the change of variables for this hyperbolic DE.

After a change of variables and some algebra, we have

$\mbox{2.3.11} \ \ u_{xx}-4u_{yy}+3u_{x}+u=0$

I am then told that the second step is a change of dependent variable

$\mbox{2.3.12} \ \ u=\exp{(\beta x)}\omega$ ,

where beta is chosen so that in the transformed equation the coefficient of $\omega_{x}$ vanishes. Differentiating 2.3.12 and substituting in 2.3.11, we obtain for $\omega$ the equation

$\omega_{xx}-4\omega_{yy}+(2\beta+3)\omega_{x}+(\beta^2+3\beta+ 1)\omega=0$

I can't obtain this equation (directly above) when I follow the directions.