# Equilibrium solution limit to differential equation

• October 27th 2013, 08:24 PM
Vishak
Equilibrium solution limit to differential equation
Can someone please help me with this one? I have found the equilibrium solutions,but I'm not sure what to do next.

Consider dx/dt = x^3 - 4x

Given a solution x(t) which satisfies the condition x(0) = 1, determine the limit(t -> infinity) of x(t).

Thanks!
• October 27th 2013, 09:00 PM
Prove It
Re: Equilibrium solution limit to differential equation
Well what did you get for your solution?
• October 27th 2013, 09:29 PM
Vishak
Re: Equilibrium solution limit to differential equation
I got the equilibrium solutions x = -2 , 2 and 0. Out out these only x = 0 is stable. Not sure where to go from here though...
• October 27th 2013, 09:45 PM
Prove It
Re: Equilibrium solution limit to differential equation
I meant, what did you get for the solution to your DE? Hint: It's separable and can be solved using partial fractions.
• October 28th 2013, 12:09 PM
HallsofIvy
Re: Equilibrium solution limit to differential equation
Have you never taken a Calculus class? This is, as Prove It said, separable- it can be "separated" as
$\frac{dx}{x^3- 4x}= \frac{dx}{x(x- 2)(x+ 2)}= dt$
Integrate!
• October 29th 2013, 04:34 PM
Vishak
Re: Equilibrium solution limit to differential equation
Thank you HallsofIvy, I got the answer.

Actually the reason I asked this question was because in the worked solutions my professor had just written:

"If x(0) = 1 then the limit = 0"

So I assumed there had to be a simpler way than actually solving the ODE. I was thinking maybe...would looking at the direction field suffice in this case?