ordinary differential equations – Analytic method for ODE problem

I am studying on a drag force ODE. My question is:

Is there any analytic method to solve $$frac{dv}{dt}+alpha v^n=g\ n in(1,2)$$ It is somehow look like Bernoulli Differential Equations $$y’ + pleft( x right)y = qleft( x right){y^n}$$ but not exactly. It is originated from falling body equation with air drag force.$$mg-kv^n=ma$$ or $$mg-kv^n=mfrac{dv}{dt}$$ that divide both sides by $$m$$ and rewrite as $$frac{dv}{dt}+frac{k}{m}v^n=g$$,$$g=9.8 frac{m}{s^2}$$ for the case of $$n=1$$ it is easy to solve $$v=gfrac mk (1-e^{-frac kmt})$$ but how to do it for the case of $$n=frac 32$$ or $$n=2$$
Realy I got to struggle with special two cases… can someone help me? I want to solve to at least for $$n=2$$ to compare with the solution that I got by numerical solutions.