Position as a function of speed

[Johnny Blade]

Homework Statement

There's an object with mass $$m$$ in movement in the horizontal axes. There's a force $$\textbf{P}$$ of constant power acting on the object. Another force is the air drag which has the magnitude of $$\beta m v^{2}$$. I need to find the position $$x$$ as a function of the speed $$v$$.

Homework Equations

$$\textbf{P} = \vec{F} \cdot \vec{v} = Fv$$ because the vectors are parallel

$$\Rightarrow F = \frac{\textbf{P}}{v}$$

$$\left|\vec{f}\right| = \beta m v^{2}$$

$$F = ma$$

The Attempt at a Solution

With these equation I plug them in $$F = ma$$ and I get $$\frac{\textbf{P}}{mv}-\beta v^{2}=\frac{dv}{dt}$$ then by multiplying by $$\frac{dx}{dx}$$ I got $$\frac{\textbf{P}}{mv}-\beta v^{2}=v\frac{dv}{dx}$$.

Then I don't know how to solve this. Or perhaps there's an easier way to this problem?

 

[rl.bhat]

P/mv - βv^2 = v*dv/dx.
Multiply by v on both the side. You get
P/m -β*v^3 = v^2*dv/dx.
So
dx = v^2*dv/( P/m -β*v^3 )
Now find the integration to find x.