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Consider the two-body equation of motion in vector form

$$

\ddot{\mathbf{r}} = -\mu\frac{\mathbf{r}}{r^3}.

$$

Show that the $f$ and $g$ functions defined by

$$

\mathbf{r} = f\mathbf{r}_0 + g\mathbf{v}_0

$$

satisfy

$$

\ddot{f} = -\mu\frac{f}{r^3},\quad \ddot{g} = -\mu\frac{g}{r^3}

$$

for arbitrary $\mathbf{r}_0$ and $\mathbf{v}_0$.