reduce PDE to normal form

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. $\displaystyle y^2u_{xx}+2xyu_{xy}+x^2u_{yy}=0$

2. $\displaystyle u_{xx}-2xyu_{xy}=0$, $\displaystyle (x \ne 0)$

Thanks for your help.

Re: reduce PDE to normal form

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

$\displaystyle x r_x + yr_y = 0$.

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

$\displaystyle r = y/x, s = y$

or

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

Both will lead to your standard form.