Classification of solution of non-linear first order partial differential equation

Question

A solution of the PDE

$xu_x+yu_y+(u_x)^2+(u_y)^2-u=0$

represents

1. an ellipse in the $x$-$y$ plane.

2. an ellipsoid in the $xyu$ plane.

3. a parabola in the $u$-$x$ plane.

4. a hyperbola in the $u$-$y$ plane.

I don't have any idea to start this question.

If anyone can suggest some reference to read for this then it will be really helpful.

Thanks in advance!

Answer 1

Just looking at the form of the PDE draw us to think that a solution might be a polynomial of second order. Moreover there is a symetry with $x$ and $y$. So, we can try, if by chance, a simple form would be $u(x,y)=a(x^2+y^2)+bxy$

Puting it into the PDE, by identification we observe that the equality is satisfied with $a=-\frac{1}{8}$ and $b=\pm\frac{1}{4}$

So, we found that $u(x,y)=-\frac{1}{8}(x^2+y^2)\pm\frac{1}{4}xy$ are solutions of the PDE.

I let you conclude what represents the equation : $$u+\frac{1}{8}(x^2+y^2)-\frac{1}{4}xy=0$$ or the equation : $$u+\frac{1}{8}(x^2+y^2)+\frac{1}{4}xy=0$$