Partial Differntial problem Cauchy

[arrow27]

Find surface of

$\begin{array}{l}
\text{Problem Cauchy} \\
{a^2} \cdot {x_2} \cdot u \cdot {u_{{x_1}}} + {b^2} \cdot {x_1} \cdot u \cdot {u_{{x_2}}} = 2{c^2}{x_1}{x_2}{\rm{ }} \\
\end{array}$
The partial differntial equation passes through

${\rm{ C: = \{ }}\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,{x_3} = u({x_1},{x_2}) = 0\} \\
\\ $$a,b,c$ nonzero constants

 

[Mathwebster]

First. I think you should be consistent with your notation. Either use \(\displaystyle x\) and \(\displaystyle y\) or \(\displaystyle x_1\) and \(\displaystyle x_2\) but not both. It's confusing.

Second, as you have the boundary of an ellipse, have you thought of introducing new coordinates

\(\displaystyle x = a r \cos \theta, y = b r \sin \theta?\)

 

[arrow27]

Ιn C are x1,y1.