I'm stuck on how to approach this question, any tips/solutions would be amazing.
Consider the initial value problem $$\frac{d^2\mathbf{x}}{dt^2}=-\frac{GM}{r^3}\mathbf{x}$$ with $\mathbf{x}(0)=\mathbf{x}_0$ and $\frac{d\mathbf{x}}{dt}(0)=\mathbf{v}_0$. For which values of $\mathbf{x}_0$ and $\mathbf{v}_0$ does this problem have a unique solution $\mathbf{x}(t)$ for sufficiently small $t$?
Note that $r=|\mathbf{x}|=\sqrt{x_1^2+x_2^2+x_3^2}$ and $G$ and $M$ are constants.
Thanks in advance!