reduce PDE to normal form

**ellenu485** · September 3rd 2011, 09:02 PM

Hi, I am having difficulty doing these 2 questions. I tried the change of coordinates but that got very messy and got me nowhere.

Reduce to canonical form:
1. $y^2u_{xx}+2xyu_{xy}+x^2u_{yy}=0$
2. $u_{xx}-2xyu_{xy}=0$ , $(x \ne 0)$

Thanks for your help.

**Danny** · September 4th 2011, 07:08 AM

For the first one, since $B^2 - 4AC = 0$ , then the PDE is parabolic and the normal or standard form is $u_{ss} + l.o.t.s. = 0$ . Substitute the change of variables in and collect about the like higher or terms (i.e. $u_{rr}, u_{rs}, u_{ss}$ ). Setting the coefficient of $u_{rr}$ and $u_{rs} = 0$ gives

$x r_x + yr_y = 0$ .

This we solve giving $r = R(y/x).$ Now $R$ can be as anything involving $y/x$ and $s$ can be chosen as really anything provided that $r_xs_y - r_ys_x \ne 0$ . For example,

$r = y/x, s = y$

or

$r = \ln x - \ln y, s = \ln y$ .

Both will lead to your standard form.

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