Nonlinear second order ODE describing a force field

  • #1
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 [itex]F=-\frac{dU}{dx}[/itex] in one variable,

[itex]F(x)=-\frac{a}{b}+\frac{ba}{x^{2}}[/itex]

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

Homework Equations



Dividing by mass and multiplying by x^2:

[itex]x^2\frac{d^2x}{dt^2}=-\frac{ax^2}{mb}+\frac{ba}{m}[/itex]

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


The Attempt at a Solution



[itex]x=y, \frac{-a}{mb}=b, \frac{ba}{m}=k, y'=u[/itex]

[itex]y^2y''=by^2+k[/itex]

I tried to eliminate the y'':

[itex]y'dt=udt[/itex]

[itex]\int{y'dt}=\int{udt}[/itex]

[itex]y=ut+C[/itex]


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.
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
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hi thetasaurus! :smile:

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 :wink:)
 

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