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Liner differentials of order n, Kernel

  1. Apr 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Verify that the given function is in the kernel of L.

    L = x2D2 + 2xD - 2

    2. Relevant equations

    3. The attempt at a solution

    I took the first and 2nd derivative of y(x), and got
    y'(x)= -2x-3
    y''(x)= 6x-4

    Then plugged it into L (and a little simplifying) and got

    L(y) = 6x-2+2x-1-2

    I think I'm supposed to plug it in, and verify that it's equal to zero, but it's not coming out right.

    Any obvious mistakes? Or wrong direction all together?
  2. jcsd
  3. Apr 26, 2009 #2


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    How does 2xDy become 2x^(-1)?? And the -2 isn't just a -2. L is operating on y. What should it be?
  4. Apr 26, 2009 #3
    Bah, forgot about the -2 part. It's actually -2y, correct? So the last term would be -2y, or -2x-2.

    And as I was typing out how I came up with 2xD, I realized I substituted just y into D, and not y' :blushing:

    With the correct substitutions, I came up with:

    L = x2*6x-2 + 2x*-2x-3 - 2x-2

    = 6x-2 - 4x-2 - 2x-2

    = 0

    :tongue: Once again, thanks Dick.
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