Type of PDE help

**ellenu485** · August 30th 2011, 01:50 PM

Hi, I am having trouble with these questions and any help would be appreciated.

Q1) Find the regions in the xy plane where the equation is elliptic,hyperbolic, or parabolic. Also sketch them.
$(1+x)yu_{xx}+2xyu_{xy}-xyu_{yy}=0$

Q2) Same as q1 but the equation is:
$(1+x)u_{xx}+2xyu_{xy}-y^2 u_{yy}=0$

Thanks

**Danny** · August 31st 2011, 04:42 AM

If you let $A = 1+x, B = 2xy$ and $C = -xy$ , then you go to consider when $B^2-4AC >0, = 0, <0$ . This will tell you when the PDE is hyperbolic, parabolic or elliptic.

**ellenu485** · September 2nd 2011, 03:07 PM

Originally Posted by Danny

If you let $A = 1+x, B = 2xy$ and $C = -xy$ , then you go to consider when $B^2-4AC >0, = 0, <0$ . This will tell you when the PDE is hyperbolic, parabolic or elliptic.

Thanks Danny, but shouldn't $A=(1+x)y$ ? So I get $B^2-4AC=4xy^2(2x+1)$ .

So for:
Parabolic:
$4xy^2(2x+1)=0$
$x=0$ , or $y=0$ , or $x=1/2$

Hyperbolic:
$4xy^2(2x+1)>0$
$2x+1>0$
$x> -1/2$

Elliptic:
$4xy^2(2x+1)<0$
$x< -1/2$

Is this correct?

And for 2) I get $B^2-4AC=4y^2(x^2+x+1)$
Parabolic: $y=0$ since $x^2+x+1 \ne 0$
Hyperbolic: $x^2+x+1>0$ (the entire graph)
Elliptic: $x^2+x+1<0$ (which is nothing)

Thanks

**ellenu485** · September 3rd 2011, 09:04 PM

EDIT: sorry, thought I made a mistake.

Can anyone confirm that my above working is correct?

**Danny** · September 4th 2011, 06:18 AM

Originally Posted by ellenu485

Thanks Danny, but shouldn't $A=(1+x)y$ ? So I get $B^2-4AC=4xy^2(2x+1)$ .

So for:
Parabolic:
$4xy^2(2x+1)=0$
$x=0$ , or $y=0$ , or $x=1/2$

Hyperbolic:
$4xy^2(2x+1)>0$
$2x+1>0$
$x> -1/2$

Elliptic:
$4xy^2(2x+1)<0$
$x< -1/2$

Is this correct?

And for 2) I get $B^2-4AC=4y^2(x^2+x+1)$
Parabolic: $y=0$ since $x^2+x+1 \ne 0$
Hyperbolic: $x^2+x+1>0$ (the entire graph)
Elliptic: $x^2+x+1<0$ (which is nothing)

Thanks

Yep - you're right. I missed the $y$ on $A$ . As for you're workings. For the hyperbolic case your want $y \ne0, x(2x+1) > 0$ . So in addition to want you have we also have $x > 0$ .

Your second part looks good.

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