Super-hard differential equation in classical mechanics problem

[snaek]

Homework Statement

A particle of mass m moves in the following (repulsive) field
U(x) = α/x², α > 0,
with α a constant parameter. Determine the (unique) trajectory of the particle, x(t), corresponding to the initial conditions of the form

x(t0) = x0 > 0,

x'(t0) = $\sqrt{\frac{2}{m}(E-\frac{α}{(x0)^2})}$

Homework Equations

^

The Attempt at a Solution

mx''(t) = -d/dx U(x)
= - (2α/x³)
= 2α/x³

=> x''(t) = 2α/mx³

Would anyone please be able to point me on the right track to solving this stupid DE? It's really starting to mess with my mind now!

 

[Chopin]

Welcome to PF!

I think case (2) on the page http://www.sosmath.com/diffeq/second/nonlineareq/nonlineareq.html corresponds to your equation, right? If so, that should help you find the answer.

 

[Chestermiller]

You can solve your "stupid" DE by multiplying both sides of the equation by x'. This will give you exact differentials on both sides.

 

[vela]

=> x''(t) = 2α/mx³

Would anyone please be able to point me on the right track to solving this stupid DE? It's really starting to mess with my mind now!

The DE always speaks well of you.

You can solve your "stupid" DE by multiplying both sides of the equation by x'. This will give you exact differentials on both sides.

This is a good trick to remember.