Examining a differential equation of motion

[JordanGo]

Homework Statement

Examine the equation

x¨ + β(x^2 − 1)x˙ + x = 0

Explain qualitatively the motion in three cases: |x| < 1, |x| > 1, and |x| = 1. In
each case, do you expect the motion to be bounded our unbounded?
Define the energy of the system as E ≡ 1/2 (x^2 + v^2). Show that the time rate of
change of the energy is β(1 − x^2)v^2
Examine for each of the three cases in which case is energy a conserved
quantity? What happens in the other cases?

Homework Equations

x¨ + β(x^2 − 1)x˙ + x = 0

The Attempt at a Solution

I have no idea how to go about this question, the differential equation is not solvable. How do you examine it then?

 

[Mindscrape]

You don't necessarily need a solution to an ode to examine its stability. However, I suppose it may be tough to intuitively be able to describe the motion very well. I mean, besides saying it's a spring with a nonlinear damping term. You could always solve it numerically. Universities always have licenses to MATLAB and mathematica. If you don't have one of those you can use octave or another open source program.

 

[Kurt Peek]

Hi JordanGo,

As Mindscrape noted, your ODE corresponds to a damped harmonic oscillator with a nonlinear damping term. To be specific, the damping term is the coefficient $\beta(x^2-1)$. If this coefficient is positive, then the damping force will be in the opposite direction of the speed $\dot{x}$, and the mass will be "slowed down" or "damped". Conversely, if the coefficient is negative, the mass will be "sped up" and its motion "amplified". Finally, if the coefficient is zero, the motion is undamped. It is easy to see that the coefficient is positive if $|x|>1$.

As for the second part of the question, this is just a matter of filling in $v=\dot{x}$ into the expression for $E$, differentiating using the chain rule, and substituting for $\ddot{x}$ in the resulting expression from the original ODE. (Incidentally, this expression also shows that the system's energy increases if $|x|<1$).

Hope this helps!


Kurt Peek

 

[JordanGo]

Thank you guys, now I understand what to do! I really appreciate it!