Linear Systems (differential equations)

[EV33]

Homework Statement

I don't have a homework question. I have more of a general question.

Let's say we have a solution...

x=C1*[a1 a2]^{T}*e^{lamda1*t} + C2*[b1 b2]^{T}*e^{lamda2*t}

As t approaches infinity how do you determine what the solution does?

Homework Equations

The Attempt at a Solution

My guess is that a1,a2,b1,b2 don't matter.

If lamda1>0,lamda2>0 , and lamda1>lamda2 then as t approaches infinity then x will look more like the first term. As t approaches negative infinity the solution will look more like the second term.

If lamda1>0 and lamda2<0 then as t approaches infinity the solution will look more like first term. When t approaches negative infinity the solution will look more like the second term.

The coefficients (C1,C2) help determine whether the solution will approach negative infinity or infinity as t approaches one or the other.

So if the dominant term has a postive lamda and a negative coefficient then the solution will approach negative infinity as t approaches infinity?

So do I have it right?

Thank you for your time.

 

[EV33]

Just to clarify a few things.

T=transpose
t=variable
lamda1,lamda2,c1,c2,a1,a2,b1,b2= constants

 

[Mark44]

It depends on whether lambda₁ and lambda₂ (the correct English spelling) are positive or negative.

Your solution x is the sum of two vectors. Let's look at each of them separately. If \lambda_1 > 0, then e^{\lambda_1 t} approaches infinity as t approaches infinity. This means that the first vector in the sum is growing longer as t gets large.

If \lambda_1 < 0, then e^{\lambda_1 t} approaches zero as t approaches infinity. This means that the first vector in the sum is getting shorter as t gets large.

The same kind of analysis can be done for \lambda_2, and its effect on the second vector.

 

[EV33]

Thank you that answers part of my question. I am trying to look at x as a whole though to see how x changes as t goes to infinity or negative infinity. I am curious if my thoughts for how the system as a whole is correct. In my first post I give a few specific examples. Were my assumptions correct?

 

[Mark44]

What you said is mostly correct. The part that is incorrect is what you said about the solutions approaching infinity or negative infinity. The solution is a sum of two vectors, not a number.