Uniqueness of solutions to the ODE

Question

Cnsider the following ODE: $\frac{dx}{dt}=-x^{\frac{1}{3}}$

If two solutions x(t) and y(t) are defined on the interval $(t_{0}-a,t_{0}+a)$ and $x(t_{0})=y(t_{0})$, then show that $x(t)=y(t)$ $\forall t_{0}\le t$

In the exercise they give the hint of caclculating the derivative of $(x(t)-y(t))^2$, but i don't see how that is any helpful

Answer 1

I don't think this is true.

Let $y=0$.

Let $x(t) = \sqrt[{3 \over 2}]{-{2 \over 3} t}$ for $t < 0$ and zero otherwise.

Both solve the equation and have the same initial condition at $t=0$.