I am trying to find a solution to Newton's equation of motion $ \boldsymbol{F} = m \boldsymbol{\ddot{r}} $ assuming a constant force $ \boldsymbol{F} $ but accounting for kinetic friction which is a force of constant magnitude $ \mu_{\mathrm{k}}\, g \, m $ and opposing the velocity.
$$ \boldsymbol{F} - \mu_{\mathrm{k}}\, g \, m \frac{\boldsymbol{\dot{r}}}{\| \boldsymbol{\dot{r}} \|} = m \boldsymbol{\ddot{r}} $$
I tried approximating $ \| \boldsymbol{\dot{r}} \| $ with $ \sqrt{\boldsymbol{\dot{r}}^{2} + \epsilon} $ but failed to find a solution. Is there a closed form solution? If not, is there an easy way to prove this? How would one construct a approximation of the solution?