1

The problem

Consider the equation:

$$xu_x - yu_y = 0.$$

Let $u$ and $v$ be functions in $\mathcal{C}^1$ that solve the equation above and knowing that $u(1, y) = v(1, y)$, determine the largest subset $A \subset \mathbb{R}^2$ where $u$ and $v$ coincide.

My attempt

I know that the characteristic of this equation will be $C = | xy |$. But I don’t know how to determine the largest region where the solution is unique. How should I proceed?

Rócherz
  • 3,976
Occhima
  • 211

1 Answers1

0

If $u(1,y)=v(1,y)$ for all $y\in\mathbb{R}$, then $u$ and $v$ coincide for all $(x,y)\in\mathbb{R}^2$.

Proof: Suppose, by contradiction, that there is $(a,b)\in\mathbb{R}^2$ such that $u(a,b)\neq v(a,b)$. Since the general solution to $xu_x-yu_y=0$ is $u(x,y)=f(xy)$, where $f$ is an arbitrary $\mathcal{C}^1$ function, that implies $u(1,ab)=u(a,b)\neq v(a,b)=v(1,ab)$, which contradicts the hypothesis that $u(1,y)=v(1,y)$ for all $y\in\mathbb{R}$.

Gonçalo
  • 9,312