The equation of motion (in the center of mass frame) due only to gravitational forces between two point masses is:
$$\frac{d^2r}{dt^2} = -\frac{GM}{r^2}$$
How does the equation get modified when a repulsive force due to vacuum energy/dark energy is included?
We want the Newtonian limit of the Einstein Field equations for nonzero vacuum energy(=cosmological constant). As $\rho_\mathrm{vac}=\Lambda/4\pi G$ is a mass(=energy) density, Poisson equation is $$ \Delta\Phi=4\pi G\rho(\boldsymbol r)-\Lambda \tag{1} $$
If we assume spherical symmetry, and point-like source $\rho\sim\delta(\boldsymbol r)$, the grativational potential that solves $(1)$ is $$ \Phi(r)=-\frac{GM}{r}-\frac{1}{6}\Lambda r^2\tag{2} $$ so that the gravitational acceleration is given by $$ g=-\partial_r\Phi=-\frac{GM}{r^2}+\frac{1}{3}\Lambda r\tag{3} $$
You get an extra term that increases with r:
$$a = -\frac{G\cdot M}{r^2} + j\cdot r$$
with j as the repulsive component.
