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

  1. Feb 23, 2013 #1
    I need to show the intervals where the wronskian is linearly independent and linearly dependent.. I don't know how to do that.. Here's what I have:

    2mqqSev.jpg
     
  2. jcsd
  3. Feb 23, 2013 #2

    Dick

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

    Odd question. The functions are linearly independent on an interval if the Wronskian is nonzero ANYWHERE in the interval.
     
  4. Feb 23, 2013 #3
    Re: D.E. Wronskian Method: Stuck trying to show L.I. and L.D. interval

    So what is the next procedural step?
     
  5. Feb 23, 2013 #4

    Dick

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

    Procedure is done. Next step is to think about it. The wronskian vanishes only at 0 and 2/7. Can you think of any intervals where the wronskian vanishes? Don't think too hard. I don't think the question makes much sense.
     
  6. Feb 23, 2013 #5

    LCKurtz

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

    Did you understand Dick's reply? There is no positive length interval on which those functions are linearly dependent. Once you see that, there's nothing left to do. Unless you want to consider a single point as an interval.

    [Edit] I didn't see Dick's second reply, sorry.
     
  7. Feb 23, 2013 #6
    Re: D.E. Wronskian Method: Stuck trying to show L.I. and L.D. interval

    No, I didn't understand it. I need to see some sort of visual, physical representation of what's going on. :/ It's all just made up vocabulary for made up concepts at this point. All I know is that if you can solve for x, you call it the vocabulary word, Linearly dependent. If not, you call it linearly independent.
     
  8. Feb 23, 2013 #7

    Dick

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

    That's ok. An interval consisting of a single point is also pretty uninteresting. That makes the two functions just numbers. Always linearly dependent on single point intervals no matter what the wronskian. I can't figure out whether the person that wrote the question doesn't understand wronskians or is just trying to make a 'trick' question.
     
  9. Feb 23, 2013 #8

    LCKurtz

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

    I was wondering the same thing. This sort of problem usually comes up where the point is to show the student how the Wronskian vs. linear dependence differs for two solutions of a 2nd order linear DE vs. two arbitrary functions. But this exercise completely misses the target if indeed that was the target.
     
  10. Feb 23, 2013 #9

    Dick

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

    Two functions are said to be linearly dependent on an interval if there are two constants a and b not both 0 such that a*f1(x)+b*f2(x)=0 on the whole interval. If not then they are linearly independent. That's all. What's important here is that you learn what the wronskian has to do with that. I told you but also see here. http://en.wikipedia.org/wiki/Wronskian#The_Wronskian_and_linear_independence
     
  11. Feb 23, 2013 #10
    Re: D.E. Wronskian Method: Stuck trying to show L.I. and L.D. interval

    So what should I write to answer part a and part b?
     
  12. Feb 23, 2013 #11

    Dick

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

    Given what you've been told, tell us what you think. Take the interval (1,2). Linearly dependent or independent? Why? How about (-100,100)?
     
  13. Feb 23, 2013 #12
    Re: D.E. Wronskian Method: Stuck trying to show L.I. and L.D. interval

    In my mind, there's no connection for either of the phrases, "linearly independent/dependent" to actual concepts. "Two functions are said to be linearly dependent on an interval if there are two constants a and b not both 0 such that a*f1(x)+b*f2(x)=0 on the whole interval." doesn't mean anything to me either. What interval? What constants?

    What would help is a heavily generalized example/explanation that goes like this, "Some guy was doing ____ and he needed to know _____ so the wronskian test was invented to tell him _____ aspect about _____."
     
  14. Feb 23, 2013 #13

    Dick

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

    "Some guy was solving a third order differential equation and got three solutions, say f1(x), f2(x) and f3(x). This guy knows that if the three solutions are linearly independent then ANY solution of the differential equation can be written as a*f1(x)+b*f2(x)+c*f3(x) for arbitrary constants a,b and c, and he has a general solution. If not, then he doesn't have the complete solution. So he checks the wronskian. If there is a point where the wronskian is nonzero, then he's all done. If it's identically zero, then the guy should suspect he may have missed a solution."
     
  15. Feb 23, 2013 #14
    Re: D.E. Wronskian Method: Stuck trying to show L.I. and L.D. interval

    1. What's another way to say "linearly independent"? What are synonyms of the phrase? If you were making up a phrase to describe whatever linearly independent describes, what are some alternate names you would come up with for it?
    2. "If there is a point where the wronskian is nonzero" what does that mean? Are you trying to say, after you set the wronskian equal to zero and solve it for x, if there are values for x the wronskian is nonzero?
    3. "If it's identically zero" What does that mean?
     
  16. Feb 24, 2013 #15

    Dick

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

    1. f1=x, f2=x^2 and f3=3x-2x^2 are linearly dependent, because f3=3*f1-2*f2. One of the functions can be expressed in terms of the others in a linear way. I think 'linearly dependent' sums it up pretty well.

    2. Your wronskian is nonzero at x=1, x=2, x=3, x=pi, x=(-1), x=0.3182, etc etc. The only places it's zero are x=0 and x=2/7. You solved it. I'm not sure what you are asking.

    3. Work out the wronskian of x, x^2 and 3x-2x^2 if you want to see what "identically zero" means.
     
  17. Feb 24, 2013 #16
    Re: D.E. Wronskian Method: Stuck trying to show L.I. and L.D. interval

    1. So it would seem that a more intuitive definition of 'linearly dependent' would be that "one of the functions can be made using the other functions" or, "the other functions can be substituted into the last function" or "the last function can be comprised of the other functions" or something to that effect followed by an example. That tells me a LOT more than the 'standard' definition does.

    2. so it's nonzero everywhere except for 0 and 2/7? If so, that's simple enough. What does that have to do with anything though? What does that actually tell me?

    3. When you work that out, you just get 0

    So to summarize, when solving a D.E., 2nd order or higher, after you get your answer, you use the Wronskian test to see if it's right? For the Wronskian test, you set the result equal to zero and solve for x. If you can find any values that solve for x then it gets labeled "linearly dependent" and that tells you what... that the answer is correct?? If you can't solve for x then it gets labeled as "linearly independent". (I know what linearly independent and dependent mean now, but I don't know how/why it's relevant to anything.) Finally, if you get 0 right off the bat when doing the wronskian, you messed up somewhere when solving the D.E.
     
  18. Feb 24, 2013 #17

    Dick

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

    It tells you the functions are linearly independent. You don't even have to solve for x to know your wronskian is nonzero. Just put in x=1. You get -20e^(-3)(5). That's nonzero. So the functions are linearly independent on any interval containing x=1.
     
  19. Feb 24, 2013 #18
    Re: D.E. Wronskian Method: Stuck trying to show L.I. and L.D. interval

    But I wrote down that they are linearly dependent which I based off of my notes. I don't know what to put for part A and part B. (2/7, inf)? (-inf, 0)? (0, 2/7)?

    This supposed to be a test to see if your answer is right but getting any kind of meaning out of the results of the test is where I don't follow.. So you solve for x and you either get one or more x values or you don't. Where's the connection between that, and knowing if your general solution to the D.E. you were trying to solve is correct? If you get results for when solving for x, that means the solutions are linearly dependent. "If there is a point where the wronskian is nonzero, then he's all done" It's nonzero everywhere except 0 and 2/7 so that means the answer is right.... I guess, but that doesn't tell me what to put for part A and B
     
  20. Feb 24, 2013 #19

    Dick

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

    Start with your guess of (2/7,inf) as an answer for A. Are the functions linearly independent on (2/7,inf)? Why?
     
  21. Feb 24, 2013 #20
    Re: D.E. Wronskian Method: Stuck trying to show L.I. and L.D. interval

    I don't have any idea how to answer your question. I have no idea how to tell if the functions are L.I. or L.D. on a certain interval. Just guessing, I plugged in 2/7 into y1 and y2. That gave me 5e^(4/7) and 16/49*e^(-10/7) ....Is that supposed to mean something?
     
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