Stable, Neutrally Stable, and Unstable Systems

[jdawg]

Homework Statement

I'm having difficulty knowing how to determine if a given differential equation represents a stable, neutrally stable, or an unstable system. I was given a hint to focus on the homogeneous solution, so is the particular solution not important for determining if the differential equation represents a stable, neutrally stable, or an unstable system?

I think I'm supposed to plug in infinity to the solution to the DE, and if x(t) ends up going to infinity then it will be unstable? What result would make the system neutrally stable or unstable?

Also, when trying to solve for the particular solution, I was told to guess either a constant, x, or some polynomial depending on what the forcing function of the DE was. But looking at this example, I must be guessing incorrectly because it isn't making sense:

dx/dt = 3
the homogeneous solution:
me^mt=0 therefore m = 0:
x_h(t)=Ce^(0)t=C

So then when you try to solve for the particular solution, your forcing function is a constant, so you guess that it is equal to a constant:

x_p=D
But the derivative of this is zero! so then when you go to plug the derivative back into your original equation, you get 0=3, which doesn't make sense. Maybe I'm missing something!

I hope my questions aren't too unorganized or confusing. Thanks for any help!

Homework Equations

The Attempt at a Solution

 

[scottdave]

What methods have you learned, so far? There is a simple Calculus method for this particular one (dx/dt = 3).

 

[jdawg]

Haha oh wow, I can't believe I didn't think of doing separation of variables with this one. I've already taken DE, but that was about 2 years ago. In the class I'm taking now we only use the characteristic method and Laplace Transforms.

Does the characteristic method not work for this problem?

 

[scottdave]

Haha oh wow, I can't believe I didn't think of doing separation of variables with this one. I've already taken DE, but that was about 2 years ago. In the class I'm taking now we only use the characteristic method and Laplace Transforms.

Does the characteristic method not work for this problem?

Ha. It's been longer for me. I'm going to have to re-read about the characteristic.

Laplace should be pretty easy. Laplace can help with the stability question. Have you learned about poles and zeros? The location of the poles will determine stability of a system.

 

[jdawg]

Yeah I'm not sure if there is some sort of condition that has to be met for you to be able to use the characteristic method? Cause I had two homework problems where it didn't seem to work. The other problem I had was a second order linear DE so I don't think I could use separation of variables with it :(

We just started relearning Laplace yesterday, so I'm pretty rusty with it. Poles and zeros don't sound familiar at all to me!