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Properties of the D operator

  1. Jan 30, 2012 #1

    sharks

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    The problem statement, all variables and given/known data
    http://s1.ipicture.ru/uploads/20120130/fcGLnUw5.png

    The attempt at a solution
    I have been trying to understand how to obtain the R.H.S. of each property from its L.H.S. but i can't find how, although i know that it's somehow related to differentiating the L.H.S. I am having a hard time to prove these properties, starting with the first one.
     
  2. jcsd
  3. Jan 30, 2012 #2

    I like Serena

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    What does the L function stand for?

    If we disregard the L, or consider it the identity, you've got the differentiation rules for a couple of standard functions, combined with the application of the chain rule.
     
  4. Jan 30, 2012 #3

    sharks

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    In my notes, L(D) is a function of the D operator.
    Symbolically, a differential equation can be written in the form: L(D)y=f(x)
     
  5. Jan 30, 2012 #4

    I like Serena

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    The question remains what kind of function.
    It doesn't seem to do anything useful.

    Your differential equation L(D)y=f(x) would be the same as y'=f(x).
    Or with other notations: ##D_x y=f(x)##, or ##{dy \over dx}=f(x)##.
     
  6. Jan 30, 2012 #5

    lanedance

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    is L just any function?
    [tex] L(D)y(x) = f(x) [/tex]

    for example when [itex] L(D)=D^2 +1 [/itex] you have
    [tex] L(D)e^ax = D(De^{ax}) +e^{ax} = (aDe^{ax})+e^{ax}=(a^2+1)e^{ax}=L(a)e^{ax} [/tex]
     
  7. Jan 30, 2012 #6

    sharks

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    There's not much more explanation about the function L in my notes.

    The Particular Integral, [itex]y_p=\frac{1}{L(D)}f(x)[/itex] and then there's a whole table of Inverse Operator Techniques. For example, [itex]y_p=\frac{1}{L(D)}ke^{ax}[/itex] gives [itex]\frac{ke^{ax}}{L(a)}[/itex], [itex]L(a)\not=0[/itex]

    It seems to me like the function L simply retains the value that has to be substituted into the function for D. For example: [itex]L(D)=5D^2+3D+1[/itex] where D=a=2 would give something like [itex]L(2)=5(2)^2+3(2)+1[/itex] but i don't know what kind of function it is.
     
    Last edited: Jan 30, 2012
  8. Jan 30, 2012 #7

    micromass

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    There must be something given for L??

    Is L linear?? A polynomial?? Given by a power series?? Continuous??
     
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