Im am dealing with this differential equation:
$$m\frac{dv}{dt}=mg-kv^2$$
where $m,g,k$ are constants.
I am able to solve this by treating this as a separable differential equation, but that method is long and tedious and there is lots of room to make mistakes. I am wondering if there is an easier method to solve this i.e laplace transform, etc?
Thanks in advance
One sort of solution to this to make it moderately neater is to write $v(t)=Cf(\lambda t)$, and choose $C,\lambda$ craftily to simplify the algebra:
$$m\frac{dv}{dt}=mg-kv^2\implies mC\lambda\cdot f'=mg-kC^2f\implies f'=\frac{g}{C\lambda}-\frac{kC}{m\lambda}f^2$$
So, we might reasonably take $\frac{g}{C\lambda}=1,\frac{kC}{m\lambda}=1\implies C=\sqrt{mg/k},\lambda=\sqrt{gk/m}$, and thereafter work with $f'=1-f^2$, which is perhaps preferable