Homework 3, due February 4, 2008.

[goodwine]

From the course text: 6.1 (if you have not already done it), 6.9-6.12, 7.4 and 7.5. If you did do 6.1 on homework 2, you don't have to do it again, but indicate that you did so that it is easy for the graders to keep track of it.

[sjensen1]

Problem 6.8 was assigned last week. Should we still work through it?

[goodwine]

Problem 6.8 was assigned last week. Should we still work through it?

Sorry, that was a mistake. I edited the assignment to remove it.

[Anon]

Professor,

If we did problem 6.1 in assignment #2, but applied the incorrect method to solve for the solution may we re-do it to obtain full credit as opposed to the credit that might have been lost by doing it in assignment #2?

[wgarraha]

Should we assume any kind of units for problem 7.4 or should we leave b, k, and m as variables?

[AlexDarr]

Both questions are about 6.11:

1) For irrational numbers (1+/- sqrt(15)), two of the eigenvalues of the matrix, is there a relationship between their eigenvectors, like there is with complex numbers?

2) Evaluating (A-lamda*I) where lamda is 1+/- sqrt(15), I got eigenvectors of [0;0;0;0] for both. I checked my math for both, and it seems right. Are two trivial eigenvectors acceptable, since the other eigenvalues yield nontrivial eigenvectors?

[AlexDarr]

Scratch that last post

[satkinso]

Should we assume any kind of units for problem 7.4 or should we leave b, k, and m as variables?

I'm actually wondering myself if, when it says, "Determine the equations of motion," it means to just find the differential equations?

[goodwine]

Should we assume any kind of units for problem 7.4 or should we leave b, k, and m as variables?

I'm actually wondering myself if, when it says, "Determine the equations of motion," it means to just find the differential equations?

Yes the equations of motion are the differential equations. Unless it says to solve them, you don't have to do so.

[gtorrisi]

for problem 7.5 when it says to write a computer program to determine an approximate numerical solution for the system, do you mean you want us to show a graph of the motion vs. time, and if so, do we need to show each mass or just the first one, or is it asking for something else?

[goodwine]

for problem 7.5 when it says to write a computer program to determine an approximate numerical solution for the system, do you mean you want us to show a graph of the motion vs. time, and if so, do we need to show each mass or just the first one, or is it asking for something else?

It's intentionally open-ended. Submit something that clearly communicates the nature of how the masses are moving. It's up to you to determine what you think is best.

[goodwine]

If we did problem 6.1 in assignment #2, but applied the incorrect method to solve for the solution may we re-do it to obtain full credit as opposed to the credit that might have been lost by doing it in assignment #2?

You can re-do it, but put a note on your homework saying that you want this one to count and not the previous one.

[goodwine]

I had a few questions on problem 7.5: How do you fit the f(t)
equation into the xsidot = a*xsi? Is it just an added on thing after
that? Secondly, in the computer program is it possible to input
everything as an array or do you need to do x1dot(1) = x1(2),
x1dot(2) = etc..? Thanks

For the first question, it is an extra term since it does not depend on any of the components of xi.

For the second question, it depends on the programming language you use, but the answer is probably that you have to input everything term by term.

[goodwine]

I am having trouble solving the differential equations with repeated eigenvalues. In the text book, in both examples the matrix (A-xI)=0 (where x = lambda), when squared or cubed or multiplied to the fourth power, always becomes a matrix of zeros. But in the homework problems, the matrices do not( they oftentimes just become greater ratios of the same number), and I can never find more than one eigenvector, or sometimes even no eigenvectors. In any case, I believe that I need m eigenvectors before I can plug them back into the given equation. Is there something that I am doing wrong?

Either you are computing the powers incorrectly or after the powers you can do a lot of row reduction. I don't remember exactly for the assigned questions, but you are not guaranteed a matrix full of zeros. You may have to do some row reduction to get m rows of zeros.