Nonlinear second order ODE describing a force field

[thetasaurus]

Not sure if this topic belongs here, but here goes.

Homework Statement

From the AP physics C 1995 test there is a problem that gives the potential energy curve U(x). With $F=-\frac{dU}{dx}$ in one variable,

$F(x)=-\frac{a}{b}+\frac{ba}{x^{2}}$

Where a and b are constants. Now I need to get x(t)

Homework Equations

Dividing by mass and multiplying by x^2:

$x^2\frac{d^2x}{dt^2}=-\frac{ax^2}{mb}+\frac{ba}{m}$

Unfortunately I do not have the skills to solve this differential equation.

The Attempt at a Solution

$x=y, \frac{-a}{mb}=b, \frac{ba}{m}=k, y'=u$

$y^2y''=by^2+k$

I tried to eliminate the y'':

$y'dt=udt$

$\int{y'dt}=\int{udt}$

$y=ut+C$And that doesn't really get me anywhere. Anyone with knowledge of nonlinear ODEs care to help? I tried Wolfram, but even with my Pro free trial it took to much computing time and never gave me a solution.

Also since this wasn't required of the problem per se, and I just want to solve this, I'm not sure what forum it should be in.

Thanks.

 

[tiny-tim]

hi thetasaurus!

try the standard trick (from the chain rule) …

d²x/dt² = v dv/dx (where v = dx/dt)

(btw, that's where 1/2 mv² comes from )