# Super-hard differential equation in classical mechanics problem

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

^

## 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!

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## Answers and Replies

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Chestermiller
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.

vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
=> 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.