Non Uniqueness - ODE Consider the ODE: $x'[t]=\left|x\right|^{p/q}$, $p,q > 0$, and share no common factors. Can someone prove that for the initial condition $x[0]=0$ and $p < q$, there are an infinite number of solutions. And if $p > q$, there is a unique solution.
Cheers 
 A: Ok, you have the right idea, but you need to do the calculation a with a little bit more care, and keep track of the constants of integration.
You're correct to separate variables, so we have
$$
\int x^{-p/q} dx = t + C \\
\frac{q}{q-p}x^{\frac{q-p}{q}} = t + C \\
x = K\left(t + C \right)^\frac{q}{q-p}
$$
where $K$ is some constant determined by $q$ and $p$, but $C$ hasn't had any restrictions placed on it yet.
Now, the thing to notice about all of these functions, if $p<q$ is that, at $t = -C$, you have $x = 0$ and $x' = 0$ (verify this both mathematically, and for intuition, graphically (w/ e.g. $x = t^{3/2}$)). So, if we translate the 'basic' solution $x = K t^\frac{q}{q-p}$ to the right, it will still be a solution of the IVP, and the derivative will even still be continuous! Specifically,
$$
x =
\begin{cases} 
0 &: 0 \leq t < C, \\
K\left(t - C \right)^\frac{q}{q-p} &: t > C
\end{cases}
$$
will be a solution for every $C > 0$. 
To see why there is a unique solution in the other case, $p>q$, appeal to the broader theory of existence and uniqueness for ODEs. You should know something about Lipschitz functions, maybe?
