• Support PF! Buy your school textbooks, materials and every day products Here!

Super-hard differential equation in classical mechanics problem

  • Thread starter snaek
  • Start date
  • #1
1
0

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) = [itex]\sqrt{\frac{2}{m}(E-\frac{α}{(x0)^2})}[/itex]


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!
 
Last edited by a moderator:

Answers and Replies

  • #3
20,134
4,209
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
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,620
1,254
=> 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.
 

Related Threads on Super-hard differential equation in classical mechanics problem

  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
837
  • Last Post
Replies
2
Views
837
  • Last Post
Replies
11
Views
2K
Replies
9
Views
2K
Replies
4
Views
2K
Top