2nd Order Linear PDE: u_xx+2u_yx+5u_yy-2u_x+u=0 - Elliptic

**bugatti79** · May 30th 2011, 11:51 AM

Folks,

I believe this 2nd order PDE is elliptic. I want to make sure new coordinates are correct before I find the canonical form.

$\displaystyle u_{xx}+2u_{yx}+5u_{yy}-2u_x+u=0 \implies \frac{dy}{dx}=1 \pm \sqrt{-4}=1 \pm 2i \therefore$

$\displaystyle y=\int 1 \pm 2i dx= x \pm i 2x +constant$

The constants are in the form $\phi(x,y)-i \psi(x,y)=constant$

I choose for my new co ordinates $s(x,y)=y-x$ and $t(x,y)=-2x$

Are s and t correct?

thanks

**Danny** · May 31st 2011, 05:45 AM

Yes - they're good. I might add that you can choose $t = 2x$ (no negative) and still be good.

**bugatti79** · May 31st 2011, 07:05 AM

Originally Posted by Danny

Yes - they're good. I might add that you can choose $t = 2x$ (no negative) and still be good.

Ok, thanks Danny. The rest is jut plug and chug stuff.

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