Some differential equations I just dont get!? 10 points?

Question

I cant solve this 1st oder differential equation. Any Help would be great.

v(dv/dz)= -g+kv²

where v is the speed of the particle, z is the vertical height above some fixed point, and g and k are constants. If v=0 when z=h, find an expression for the speed of the body as a function of height.

I know the answer is

z=h+(1/2k)log(1-(k/g)v²

I've tried but cant arrive at that answer. Please help me.

Answer 1

Try this ...

dv/dz = [-g+kv^2]/v

Now it's deparable.

v/[-g+kv^2]dv=dz

Integrate both sides

And we have our answer!

let u=v^2 du=2vdv

1/2*1/[-g+ku]*du

=1/(2k)*ln[-g+kv^2]+C=z

Check by differentiation if you want:

1/(-g+kv^2)*2vk/(2k)=v/(-g+kv^2) which is what we had on LS

Now to find C use the v=0 and z=h