Movement of a particle subject to a constance force of power

[charbon]

Homework Statement

A material point A of mass m has a rectilinear movement on the horizontal axis 0x. It is subject to the action of a constant power force P and to a force due to air resistance of $$\beta$$mv². It starts at a still position on x = 0 for t = 0 in the direction of +x. Find the expression for the x-axis in function of the velocity vx:

x = $$\frac{1}{3\beta}$$ln($$\frac{P/m}{P/m - \beta v^3}$$

Homework Equations

Using the kinetic energy theorem or Newton's second law, show that vdv/dt = P/m - $$\beta$$v³
Do not try to solve this equation, introduce this relation:
dv/dt = vdv/dx to continue

The Attempt at a Solution

W_AB = K_B-K_A = K_B = 1/2mv²
P = dW/dt = mv
$$\vec{F}$$$$\bullet$$$$\vec{v}$$ = mv
$$\vec{F}$$ = (F - $$\beta$$mv²)i
Fv - $$\beta$$mv³ = mv
av = P/m + $$\beta$$v³
vdv/dt = P/m + $$\beta$$vx³

v²dv/dx = P/m + $$\beta$$v³
dv/dx = (P/m)/v² + $$\beta$$v
dv/dx = 1/v + $$\beta$$v

This is where I'm stuck. I have hard time solving complicated differential equations. Can someone walk me through the next steps?

Thanks in advance

 

[tiny-tim]

hi charbon!

(have a beta: β )

you got down to v²dv/dx = P/m - βv³ …

now separate the variables ​