How Is Velocity Expressed as a Function of Position with Given Power Input?

[e2e8]

Power to a vehicle moving along a track is alpha*x^1/2. No dissipation. alpha is constant x is distance from beginning of the track. zero initial velocity.
What is Velocity as function of position?

P=d KE/dt
P=d(1/2 m v^2)/dt
P=1/2 m d((dx/dt)^2)/dt
P=1/2 m 2 v dv/dt
alpha x^1/2 = 1/2 m 2 v dv/dt

And that is as far as I get. After some random manipulations I was not able to yield anything useful. Some assistance or guidance would be greatly appreciated. Thank you.

e2

 

[ehild]

dv/dt=dv/dx*dx/dt = dv/dx *v.

use this in your last formula then solve the differential equation for v(x).

ehild