D.E. Wronskian Method: Stuck trying to show L.I. and L.D. intervals

[Dick]

For part A) L.I. on the intervals (-inf, 0) U (0, 2/7) U (2/7). Largest interval is (-inf, 0)
..-------B) L.D. at x=0 and x=2/7

Why, why, why? Give some sort of reason! Your best answer was in post 44. Yet you continue to yammer on about this for no reason I can think of.

 

[Jeff12341234]

The answer is either:

Possible answer #1
For part A) L.I. on the intervals (-inf, 0) U (0, 2/7) U (2/7). Largest interval is (-inf, 0)
..-------B) L.D. at x=0 and x=2/7

or:

Possible answer #2
For part A) (-inf,inf). Largest interval is (-inf, inf)
..-------B) none

The way the question is worded implies the first answer is more likely to be correct. I'll keep researching it.

 

[Dick]

The answer is either:

For part A) L.I. on the intervals (-inf, 0) U (0, 2/7) U (2/7). Largest interval is (-inf, 0)
..-------B) L.D. at x=0 and x=2/7

or:

For part A) (-inf,inf). Largest interval is (-inf, inf)
..-------B) none

The way the question is worded implies the first answer is more likely to be correct. I'll keep researching it.

Keep researching it. Because I have no idea what you are talking about.

 

[Jeff12341234]

Out of all of the examples I've looked at, I've yet to see one where they found two points for x. The wronskian either simply equals zero or it doesn't in all of the examples I've seen :/

 

[Dick]

Out of all of the examples I've looked at, I've yet to see one where they found two points for x. The wronskian either simply equals zero or it doesn't in all of the examples I've seen :/

x and x^2 have a wronskian of 2x^2-x^2=x^2. They are NOT LINEARLY DEPENDENT even though the wronskian is zero at x=0. Do you think they are?

 

[Jeff12341234]

purplemath.com implies Possible answer #1 is correct: "Here W=0 only when x=0 . Therefore x^2 and x are independent except at x=0 . " http://planetmath.org/WronskianDeterminant.html

Some guy on chegg.com says that Possible answer #1 is correct: http://media.cheggcdn.com/media/c72/c72af4bb-22a6-4f97-8c51-f964eb461e24/phpdbpdV9.png

Paul's Online notes imply that Possible answer #2 is correct but maybe not since it doesn't go into details about intervals of the answer..
"Here we know that the two functions are linearly independent and so we should get a non-zero Wronskian... The Wronskian is non-zero as we expected provided . This is not a problem. As long as the Wronskian is not identically zero for all t we are okay." http://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx

You say Possible answer #2 is correct.

This implies that Possible answer #2 is correct. "Let f and g be differentiable on [a,b]. If Wronskian W(f,g)(t0) is nonzero for some t0 in [a,b] then f and g are linearly independent on [a,b]. If f and g are linearly dependent then the Wronskian is zero for all t in [a,b]." http://ltcconline.net/greenl/courses/204/ConstantCoeff/linearIndependence.htm So that's 2 votes for #1 and 3 votes for #2 so far.

 

[Dick]

So that's 2 votes for #1 and two votes for #2 so far. That's not good :(

I really don't care much what the polling is on this problem. Answer #1 is wrong. Answer #2 is correct. I think I've explained why I think so. Why do you keep asking? If you want a consensus by majority then you should keep researching to break the tie. If you can't think about this for yourself, then that seems the only option.

 

[Jeff12341234]

I keep asking because I don't fully understand it myself. I now can reliably determine if a set of functions are L.I. or not BUT, out of the 20+ examples that I've worked, none of them are like the problem that prompted this thread. none of them returned x values like this problem does. So the part that the instructor is asking about, the intervals, I don't get. The problem is either worded really retarded or answer #2 is wrong.

 

[Dick]

I keep asking because I don't fully understand it myself. I now can reliably determine if a set of functions are L.I. or not BUT, out of the 20+ examples that I've worked, none of them are like the problem that prompted this thread. none of them returned x values like this problem does. So the part that the instructor is asking about, the intervals, I don't get. The problem is either worded really retarded or answer #2 is wrong.

The problem is certainly phrased oddly. Just believe the Paul's Online Notes version of the story.

 

[Jeff12341234]

Well we got our tests back. The correct answer for part A was (-inf, 0) but he said any of the three intervals could've been used. It was L.I. on the intervals (-inf, 0) U (0, 2/7) U (2/7, inf). I got part A correct on the test.

Part B was really weird. The answer was "any interval that contained the two points" so (-inf, inf) could've been used. I got this part wrong.

Weird stuff..

 

[Dick]

Well we got our tests back. The correct answer for part A was (-inf, 0) but he said any of the three intervals could've been used. It was L.I. on the intervals (-inf, 0) U (0, 2/7) U (2/7, inf). I got part A correct on the test.

Part B was really weird. The answer was "any interval that contained the two points" so (-inf, inf) could've been used. I got this part wrong.

Weird stuff..

I don't think the person that composed the questions and answers really understands that "linearly dependent" is different from "wronskian equals zero". The PlanetMath site seemed to have a similar delusion. Hopefully, you do. Someday you might get a test question from someone who does.