Given the initial velocity $v_0$ and angle $\theta$ of a projectile on the ground, using Newton's second law and the acceleration due to gravity $\mathbf g=\left\langle0,-g\right\rangle$, I was able to derive its position vector function:
$$ \mathbf F=m\mathbf a=m\mathbf g\implies\mathbf r(t)=\left(v_0t\cos\theta,-\frac g2t^2+v_0t\sin\theta\right). $$
I now want to introduce drag into this function. From my differential equations book, I was able to deduce that
$$ m\mathbf a+c\mathbf v=m\mathbf g, $$
Where $c$ is some scalar. Is this correct? If so, how can I go about solving for $\mathbf r$? Integrating factors make no sense to me in this context.