Curves with boundary data such that $yu_x-xu_y=0$ has/hasn't a (unique) solution

Question

Consider $yu_x-xu_y=0$. Find curves and initial conditions along these curves for which this problem has a unique solution, no solution or infinitely many solutions?

I solved the equation for $u(x,y)$ and my answer is : $$u=x^2+y^2=c$$ if $c=0$ my curves are actually two lines and for $c\neq0$ my curves are circles centered at origin.
I googled this question I found some material, none of them really were answering this question precisely. i want to solve this question in general to learn how different initial conditions lead to no sol/unique sol/infinitely many sol. I appreciate your help.

Answer 1

The general solution reads $u(x,y) = F(x^2+y^2)$, where $F$ is an arbitrary function defined over $\Bbb R_+$. The general solution can obtained by solving the Lagrange-Charpit system $$ \frac{\text d x}{y} = \frac{\text d y}{-x} = \frac{\text d u}{0} \, . $$ The characteristics are the circles $x^2+y^2 = a^2$ with radius $a>0$, along which $u = F(a^2)$ is constant.

• If we impose non-constant $u$ along the circle of radius $a$, then there is no solution, see this post.
• Conversely, if we impose a constant value $U$ of $u$ along the circle of radius $a$, then there are infinitely many functions $F$ that can match the value $U = F(a^2)$. Consider for instance the family of linear functions $F:\zeta\mapsto U + \alpha (\zeta-a^2)$ with slope $\alpha$.
• Now, if we impose $u = U$ constant along the line $y=0$, then the only function $F$ that can match this boundary data is the constant function $F = U$.

Other examples can be found here.