Solve yux - xuy = 0 , y > 0 with the condition u = y when x2 + 2y2 = 4. What is the maximal domain in y > 0 where the solution is valid and determined uniquely by its Cauchy data? Draw a sketch.
What I've done so far:
If I'm correct, I've gotten the initial set of ODEs as dy/y = dx/-x and dz/dt = 0, hopefully leading to x2 + y2 = s and z = s, and therefore u = f(x2 + y2).
I know that u = y is a tilted plane that will intersect the surface of revolution of x2 + 2y2 = 4 in a half-ellipse (y > 0). I'm not sure how to finish or answer the second part of the question, or if I'm right so far.
Thanks much in advance for any hints or direction.