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Second order linear differential operator

  1. Aug 5, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose that L is a second order linear differential operator over the interval J, that f is a function defined on J, and that the function v has the property that

    Lv = f on J

    (a) Show that if y = u + v and that Lu = 0 on J, then Ly = f on J
    (b) Show that if Ly = f on J, then y = u + v for some u such that Lu = 0 on J
    2. Relevant equations
    None that apply
    3. The attempt at a solution
    Well unfortunately this one I was unable to attempt because i am not even sure what it is asking, this professor I have tends to deviate from the book.
     
  2. jcsd
  3. Aug 5, 2009 #2
    (a) If L is linear, then L(u+v)=Lu+Lv. So if we have Lu=0, then

    Ly=L(u+v)=Lu+Lv=0+Lv=Lv=f.

    (b) Since L is linear, then there is a zero vector u with Lu=0 and 0+y=y. Choose u=0 and v=y. Then u+v=0+y=y.
     
  4. Aug 5, 2009 #3
    Thanks man I would never figure that out!
     
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