Linear Systems (differential equations)

In summary: So it cannot approach infinity or negative infinity. It can only approach a certain vector or a linear combination of the two vectors.In summary, when looking at the solution x as a whole, the behavior of x as t approaches infinity or negative infinity depends on the values of lambda1 and lambda2. If lambda1 and lambda2 are both positive, x will approach a linear combination of the two vectors in the sum. If one of the lambdas is negative, the corresponding vector in the sum will approach zero, and the other vector will determine the behavior of x. The coefficients C1 and C2 will determine the overall behavior of x as t approaches infinity or negative infinity.
  • #1
EV33
196
0

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][tex]^{T}[/tex]*e[tex]^{lamda1*t}[/tex] + C2*[b1 b2][tex]^{T}[/tex]*e[tex]^{lamda2*t}[/tex]

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.
 
Physics news on Phys.org
  • #2
Just to clarify a few things.

T=transpose
t=variable
lamda1,lamda2,c1,c2,a1,a2,b1,b2= constants
 
  • #3
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 [itex]\lambda_1[/itex] > 0, then [itex]e^{\lambda_1 t}[/itex] approaches infinity as t approaches infinity. This means that the first vector in the sum is growing longer as t gets large.

If [itex]\lambda_1[/itex] < 0, then [itex]e^{\lambda_1 t}[/itex] 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 [itex]\lambda_2[/itex], and its effect on the second vector.
 
  • #4
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?
 
  • #5
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.
 

What is a linear system of differential equations?

A linear system of differential equations is a set of equations where each equation is linear in its dependent variables and their derivatives. This means that the variables are only raised to the first power and are not multiplied together. The system also contains initial conditions, which are used to solve for the specific values of the dependent variables.

What are the key features of a linear system of differential equations?

The key features of a linear system of differential equations are that it is a system of equations, meaning there are multiple equations with multiple dependent variables, and that each equation is linear in its dependent variables and their derivatives. These features allow for the use of algebraic methods to solve the system.

How do you solve a linear system of differential equations?

The most common method for solving a linear system of differential equations is by using elimination or substitution to reduce the system to a single equation. This equation can then be solved using algebraic methods, such as factoring or the quadratic formula. Alternatively, numerical methods, such as Euler's method or Runge-Kutta methods, can also be used to solve the system.

What are some real-world applications of linear systems of differential equations?

Linear systems of differential equations are used in various fields of science and engineering to model and solve real-world problems. Some examples include analyzing the spread of diseases in a population, predicting the growth of a population, studying the behavior of electrical circuits, and modeling chemical reactions.

What is the significance of eigenvalues and eigenvectors in linear systems of differential equations?

Eigenvalues and eigenvectors play a crucial role in the analysis and solutions of linear systems of differential equations. They provide information about the stability and behavior of the system and can be used to find the general solution. Additionally, they are also used in numerical methods for solving these systems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
462
  • Calculus and Beyond Homework Help
Replies
2
Views
221
  • Calculus and Beyond Homework Help
Replies
7
Views
132
  • Calculus and Beyond Homework Help
Replies
1
Views
177
  • Calculus and Beyond Homework Help
Replies
4
Views
3K
  • Calculus and Beyond Homework Help
Replies
3
Views
444
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
586
  • Calculus and Beyond Homework Help
Replies
4
Views
887
  • Calculus and Beyond Homework Help
Replies
7
Views
509
Back
Top