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Given the PDE:

$$x u_x - y u_y = 0$$

How do I show that every solution that satisfies $u(1, y^{3}) = y^3$ is not unique?.

If we apply the method of characteristics to this PDE, we get that:

$$u(x, y) = f(x y)$$

Applying the condition $u(1, y) = y^3$ we get that $u(x, y) = x^3y^3$. But I don't know how to proceed from here, what do I need to do to prove that the solutions are not unique? Is there a graphical approach to this? Iknow it has to do with the interserction of out solution with the characteristics curves of the homogenous solution.

Any tips or advices are more than welcome

Occhima
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1 Answers1

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Let $g:\mathbb{R}\to\mathbb{R}$ be any differentiable function such that $g(0)=g'(0)=0$. Then it is straightforward to verify that $$ u(x,y):=\begin{cases} x^3y^3&\text{if $x\geq 0$,} \\ g(xy)&\text{if $x<0$,} \end{cases} $$ also satisfies $x u_x - y u_y = 0$ and $u(1,y)=y^3$.

Gonçalo
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