Super-hard differential equation in classical mechanics problem

1. Sep 17, 2012

snaek

1. The problem statement, all variables and given/known data
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) = $\sqrt{\frac{2}{m}(E-\frac{α}{(x0)^2})}$

2. Relevant equations

^

3. 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!

Last edited by a moderator: Sep 18, 2012
2. Sep 17, 2012

Chopin

3. Sep 17, 2012

Staff: Mentor

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

4. Sep 18, 2012

vela

Staff Emeritus
The DE always speaks well of you.

This is a good trick to remember.