differential equation in physics?

[John A]

hi therei have a question about vertical motion
ma=mg-kv^2
dv/dt=g-(k/m)v^2 i need to solve this to find out the time it took
the ball to reach its greatest height and this is a quadratic model and
air resistance applies.
i have to solve by seperation of variables and then partial fractions
ive done the first part v/(g-(k/m)v^2)=dt but i thought from here you
would work your way down from here to make t the subject why use partial
fractions ? i thought dt=1/2t^2+c and then sort of rearrange every
thing working your way down.if you could explain all this please or give me
some tips thanks"