Originally Posted by

**Math Major** Thank you so much for that. It helped a lot to see that step worked out. Sorry for asking again, but could you possibly take a look at what I've done here and tell me where I'm going wrong? I'm sure it's something stupid.

I'm working on solving

$\displaystyle u_{xx} + u_{xy} - 2u_{yy} = 3( u_x + 2u_y + 6x - 3y) $

Once again, the equation is hyperbolic so I found appopriate substitutions to be

$\displaystyle \xi = y - 2x $ and $\displaystyle \eta = y +x $

So, taking the derivatives I find

$\displaystyle u_x = -2u_{\xi} + u_{\eta} $

$\displaystyle u_xx = 4(u_{\xi \xi} {\color{red}{+}\,} u_{\xi \eta}) + u_{\eta \eta} $

$\displaystyle u_xy = u_{\eta \eta} -u_{\xi \eta} - 2u_{\xi \xi} $

$\displaystyle u_y = u_{\xi} + u_{\eta} $

$\displaystyle u_yy = u_{\xi \xi} + 2u_{\xi \eta} + u_{\eta \eta} $

Plugging these back into the pde, I end up with the equation (in canonical form),

$\displaystyle u_{\xi \eta} = \xi - u_{\eta} $

I tried integrating with respect to eta to get

$\displaystyle u_{\xi} = \xi * \eta - u + B'(\xi) $

For some arbitrary B. However, I can't solve this ODE. I tried using an integrating factor, but that got messy quickly. I might have made an algebraic mistake in the reduction, but I've gotten the same canonical form twice.

If someone could take a look and show me where I went awry, I'd be most apprecitive.