Being at an initial condition with zero initial velocity?

Question

Suppose we're solving an autonomous DE of the form $\dot{x} = f(x)$ (where dot is derivative w.r.t. time) with some initial condition $x(0)=x_0$ (with $x_0 \in \mathbb{R}$), What significance does it have if we also know that $\dot{x}(0) = 0$? Say this system describes the movement of a particle, does that mean the particle will remain at $x_0$ for all time since it's at that point at initial time $t=0$ with zero initial velocity, so it can't move? If so, do we need any requirements for f? Perhaps Lipschitzness for the solution to be uniquely $x(t)=x_0$?

Answer 1

Yes, if $f(x_0)=0$, then the IVP with $x(0)=x_0$ has the constant solution $x(t)=x_0$, as is easy to check.

You need the Lipschitz condition for uniqueness to exclude any other solution, like they occur for the classical example $\dot x=\sqrt{|x|}$, $x(0)=0$.