I am interested in calculating the orbit of a mass around a spherical object given the initial conditions. $\overrightarrow{r_{0}}$ is the initial position and initial velocity $\overrightarrow{r^{'}_{0}}$.
The differential equation that will be used in here would be: $$\dfrac{d^{2}\overrightarrow{r}}{dt^{2}} = -\dfrac{\overrightarrow{r}}{r^{3}}$$ Is there a way to get a closed form solution of this equation? Is there a way to solve this equation in this vector form without changing it into another coordinate system?