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.

Thanks in advance.