I posted the assignment for Week #7 on Friday afternoon, but I didn't get a chance to post about it here until now. You can find it by following the link from the Schedule page. There are a couple "home-made" additional problems that I will add on Monday afternoon.
UPDATE: I've added the additional problems to the homework page for Week #7. Try refreshing your browser if they don't show up.
UPDATE: In problem 7.4.2, you should skip part (d). Do only parts (a), (b) and (c).
12 comments:
how does the first additional problem differ from 7.4.2? Am I doing it correctly if I pretty much do the same thing? if so, can I refer to that problem to save a lot of work since it should work out the same way?
Hi Anonymous
7.4.2 deals with the general homogeneous diffeq, i.e. x' = P(t)x and your answer should depend on the values of the P matrix.
In additional problem #1 you have a constant coefficient matrix, i.e. x'=Ax, but your answer should depend on the eigenvalues and eigenvectors of A. You want to show that the Wronskian is non-zero if and only if the eigenvalues are linearly independent.
Problem 7.4.2 just tells you that the Wronskian is either exactly equal to zero for all values of t, or never equal to zero. This is not enough to show the condition you need in additional problem 1.
In fact, when solving additional problem 1, don't use the method described in problem 7.4.2 since it is a lot easier to compute the Wronskian directly.
So the short answer to your question is: Yes, you can refer to previous problems, but in this case it won't help.
When doing part 7.4.2.c, I do not see any way to simplify the expression for W beyond having an integral of p11 + p22. In the text they claim that since W is an exponential function, the theorem follows immediately. But, what if p11 + p22 = 1/t. Then we get W = kt, and W is no longer exponential. Am I missing something? Thanks.
Hi drewshaver
You are not missing anything, if p11 and p22 are constants you will get an exponential. In the case that you mention, you are right that the Wronskian will be kt, but then again p11 + p22 = 1/t is continuous on an interval that doesn't contain 0, and the Wronskian will not change signs on that interval.
Thanks for clearing that up for me Pall.
Unfortunately, I have another question about the Wronskian. I looked it up on Wikipedia and the definition there differs with that of the textbook. It seems like they are connected, but why doesn't wikipedia mention our textbook's version?
Thanks.
Hi dreshawer
The Wronskian on wikipedia is for a set of real-valued functions whereas we deal with vector valued functions.
for problem 7.6.27(b), I'm not sure why xi and its conjugate must be linearly independent (I think this is the intended justification for the first part of (b)). Does it have something to do with the fact that they are eigenvectors of different eigenvalues? Can someone point me to an appropriate place in the text? Thanks :)
So now the question is...if two complex vectors x1 and x2 are linearly dependent then that means that c1x1 = c2x2. Can c1 and c2 be complex or are they scalars?
Just kidding, apparently c1 and c2 can be complex (see p.377) but i still don't see where to go. :(
I'm not sure where this is in the text, but the proof is easy. Let's say r1,r2 are two distinct eigenvalues for a matrix A with eigenvectors v1, v2. Look at the equation
c1v1 + c2v2 = 0
and multiply by the matrix A, then we get
c1r1v1 + c2r2v2 = 0
subtract r1 times the first eqn from the second one and get
c2(r2-r1)v2 = 0
Now since r1 and r2 are distinct (r2-r1) is nonzero, so we have to have that c2 = 0. We can repeat this to show that c1 = 0.
This also works for more variables, but the mechanics of the proof are a bit more complicated.
In any case you can just use this as a known fact when solving the HWs.
Thanks for the help! That's crystal clear now. :) I still can't figure out how to read these blogger captchas though...
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