I want to find the general solution of the following vector DE: $$\dot{\mathbf v}+k|\mathbf v|\mathbf v=\mathbf g$$ where $\mathbf v=\dot{\mathbf r}$, $\mathbf g,k$ being constants. This is the equation describing the projectile motion with air resistance $\propto v^2$. In one dimension, this can be easily solved with separation of variable: it ends up with an arctanh function. But this stops working in any higher dimensions.
I can also solve the case in which resistance $\propto v$ by integrating factor.
But how to solve the equation above analytically? OR is it impossible?
PS: Here is a good answer about linear drag, but it does not address quadratic drag.