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