Trivial and Nontrivial Solutions to IVPs and the Existence and Uniqueness Theorem

Question

This is a question about first order equations.

Ex: $\frac{dx}{dt} = x^2$, $x=t^3/3+C$

If the initial value is a critical point of the function (eg: $x_0=0$), the function would have a trivial and non trivial solution: the solution $x(t) = 0$ and $x^3/3=t$.

Would this not contradict the Existence and Uniqueness Theorem because $t^2$ and $t^3/3$ are both continuous but there are two solutions to the IVP $x_0=0$? What am I missing here?

Edit: Made critical point an equilibrium point.

Answer 1

$x(t) \equiv 0$ is not a solution to $\frac{dx}{dt} = 2t$.

Answer 2

If $x'(t)=t^2$, then $x(t)=\frac{1}{3}t^3+C$.

The initial value problem

$x'(t)=t^2, x(0)=0$ has the unique solution $x(t)=\frac{1}{3}t^3.$