Homework 4, due February 22, 2012

[goodwine]

Reading: All of chapter 6.

A couple things in the exercises definitely need the reading and weren't emphasized in class. If you normally don't do the reading, this week's assignment might be a time to change that.

Exercises: 6.10, 6l.11, 6.12 and 6.17.

[goodwine]

Someone asked me:

I hope you are having a good weekend. I have a question about the last problem, 6.17. I assumed that x, xdot, and y were xi 1, xi 2, and xi 3. After converting the equations into xi dot = A xi, I found the homogeneous solution. However, I cannot find a particular solution by undetermined coefficients, and the other methods are pretty cumbersome because of the repeated eigenvalues in the homogeneous solution. Is my technique off here, because I can't pinpoint what I'm doing wrong. I am planning to come to office hours on Monday, so if you can't answer through email, that's fine.

What eigenvalues did you get? I haven't done this problem in a while, but don't recall repeated eigenvalues.

[pat]

I got -1 repeated three times, so I probably messed up the conversion of the x's and y's to xi's

[goodwine]

I got -1 repeated three times, so I probably messed up the conversion of the x's and y's to xi's

Let me know if you still have -1s after you double check.

[pat]

I went over it a couple of times and ran my matrix in matlab, and I'm still getting -1's

[goodwine]

I went over it a couple of times and ran my matrix in matlab, and I'm still getting -1's

Then you must multiply by t, or t^2 and/or t^3. There is a subtle issue with it that I did not cover in class, but is addressed in the book.

[CLillie]

In regard to this same question (6.17):

I also got an eigenvalue of -1 with multiplicity of three, but e^-t is ALSO the nonhomogeneous part of the equation, so would the assumed solution have to be of the form:

A*(t^3)e^-t + B*(t^2)e^-t + C*(t)e^-t + D*e^-t

since that solution already has multiplicity of three??


Also, do Generalized Eigenvectors work for the columns of the T matrix in undetermined coefficients, or in the general solution matrix for the Diagonalization process?

[pat]

I tried that and I'm still stuck, when I solve for the undetermined coefficients, the equations that result for the coefficients combine in a way that I can't solve for them (the matrix to find the coefficients is singular). I can't find what I'm doing wrong.

[goodwine]

In regard to this same question (6.17):

I also got an eigenvalue of -1 with multiplicity of three, but e^-t is ALSO the nonhomogeneous part of the equation, so would the assumed solution have to be of the form:

A*(t^3)e^-t + B*(t^2)e^-t + C*(t)e^-t + D*e^-t

since that solution already has multiplicity of three??


Also, do Generalized Eigenvectors work for the columns of the T matrix in undetermined coefficients, or in the general solution matrix for the Diagonalization process?

Yes to the first part.

For the second part, if you do the T^1 A T thing with generalized eigenvectors can almost diagonalize it. It generally puts it in a form where you can solve the last equation, substitute into the second, solve that, sub into the first. Sort of an upper-triangular algebraic thing. I'd encourage you to do it, but none of that is in the book. Google "Jordan Canonical Form".

[goodwine]

I tried that and I'm still stuck, when I solve for the undetermined coefficients, the equations that result for the coefficients combine in a way that I can't solve for them (the matrix to find the coefficients is singular). I can't find what I'm doing wrong.

I think I need to see what you are doing to be able to help. Would my office hours tomorrow work for you (4:00-5:30)?

[pat]

Yes, I'll be there.

[Jessie]

I tried that and I'm still stuck, when I solve for the undetermined coefficients, the equations that result for the coefficients combine in a way that I can't solve for them (the matrix to find the coefficients is singular). I can't find what I'm doing wrong.

I think I need to see what you are doing to be able to help. Would my office hours tomorrow work for you (4:00-5:30)?

I am having the same issue where i get a singular matrix. Was there some general mistake that I may be making too?

[goodwine]

I tried that and I'm still stuck, when I solve for the undetermined coefficients, the equations that result for the coefficients combine in a way that I can't solve for them (the matrix to find the coefficients is singular). I can't find what I'm doing wrong.

I think I need to see what you are doing to be able to help. Would my office hours tomorrow work for you (4:00-5:30)?

I am having the same issue where i get a singular matrix. Was there some general mistake that I may be making too?

There are two possible difficulties you may encounter when trying to solve for the coefficients. The first is one or more rows of zeros with corresponding zeros on the RHS, which really isn't a problem. It just means there isn't a unique solution, which should be expected to some extent in this problem because the form of the particular solution you have to assume may have some homogeneous components in it. While it's in the context of computing eigenvectors, see page 704 (in the appendix) of the book for a systematic approach.

If you get a row of zeros with a non-zero term on the RHS, then you made an algebra error.

[Jessie]

There are two possible difficulties you may encounter when trying to solve for the coefficients. The first is one or more rows of zeros with corresponding zeros on the RHS, which really isn't a problem. It just means there isn't a unique solution, which should be expected to some extent in this problem because the form of the particular solution you have to assume may have some homogeneous components in it. While it's in the context of computing eigenvectors, see page 704 (in the appendix) of the book for a systematic approach.

If you get a row of zeros with a non-zero term on the RHS, then you made an algebra error.

Thank you!