Equilibrium solution limit to differential equation

In summary, the conversation is about finding the limit of x(t) for a given differential equation with a specific initial condition. The solution to the DE involves finding the equilibrium solutions and using partial fractions or the Bernoulli method. The person asking for help eventually finds the answer with the help of others.
  • #1
Vishak95
19
0
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!
 
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  • #2
Well what did you get for your solution?
 
  • #3
Prove It said:
Well what did you get for your solution?

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...
 
  • #4
I meant, what did you get for your solution to the DE? Hint: It's separable and can be solved using Partial Fractions.
 
  • #5
Prove It said:
I meant, what did you get for your solution to the DE? Hint: It's separable and can be solved using Partial Fractions.

It's also Bernoulli, as an alternative.
 
  • #6
Ok, thanks guys, I got the answer :)
 

FAQ: Equilibrium solution limit to differential equation

What is an equilibrium solution to a differential equation?

An equilibrium solution to a differential equation is a constant solution where the derivative of the dependent variable is equal to zero. This means that the system is in a state of balance and is not changing over time.

How is the equilibrium solution determined for a differential equation?

The equilibrium solution can be determined by setting the derivative of the dependent variable equal to zero and solving for the independent variable. This will give the value(s) at which the system is in equilibrium.

What does the equilibrium solution tell us about the behavior of a system?

The equilibrium solution can tell us about the stability of a system. If the equilibrium solution is a stable point, the system will tend towards that point over time. If the equilibrium solution is an unstable point, the system will move away from that point over time.

Can a differential equation have more than one equilibrium solution?

Yes, a differential equation can have multiple equilibrium solutions. This can occur when the derivative of the dependent variable is equal to zero at multiple points, indicating different stable or unstable states of the system.

How can the equilibrium solution be affected by changing parameters in the differential equation?

Changing parameters in the differential equation can affect the location and stability of the equilibrium solution. For example, increasing a parameter may result in a larger or more unstable equilibrium solution, while decreasing a parameter may result in a smaller or more stable equilibrium solution.

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