
Basic Question In PDE
Let D be a region in $\displaystyle R^2 $ .
Let's denote: $\displaystyle K= \{ (xa)^2 + (yb)^2 +c a,b,c \in R \} $ .
1. Prove that there is no nontrivial first order PDE
$\displaystyle F(x,y,z,z_x ,z_y) $ such as its set of soloution in D includes all the functions in K.
2. Find two non trivial differential equations of second order which are not equivalent and that all of the functions in K are soloutions for them both.
I'm pretty sure that in 1 we need to get some kind of a contradiction from fact that a function in K can be represented as:
$\displaystyle z = \frac{1}{2} z_x ^2 + \frac{1}{2} z_y ^2 +c $ ...
Help is needed
Thanks