## A general example of non-uniqueness

Consider the initial value problem

$\frac{dy}{dx} = |x|^{p/q}$, $x(0) = 0$, where $p$ and $q$ are positive integers with no common factors.

(a) Show that there are an infinite number of solutions if $p < q$.
(b) Show that there is a unique solution if $p > q$.

I separated the equations and was trying to integrate $\frac{1}{|x|^{p/q}}$ but I couldn't figure out how to do this.

I tried to break it down by the definition of absolute value so that

(1) $\frac{dy}{dx} = x^{p/q}$, $x \geq 0$
(2) $\frac{dy}{dx} = -x^{p/q}$, $x < 0$

That way, I would show that for both cases (1) and (2), parts (a) and (b) are valid. Is this okay to do, or is there a more appropriate method to solving this problem?

Thank you very much.

Regards,
crushingyen