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Why no constant of integration for the integration factor?

  1. Dec 17, 2011 #1

    nomadreid

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    Gold Member

    In applying the usual formula for y'+py=q with u=e∫p and y=(∫uq +C)/u, the results of the indefinite integral to calculate u is without a constant. Why is this? The result is not the same.
    For example, suppose I take the problem
    dy/dx +y = x*exp(-x). Then, in the usual manner, I get a nice exp(-x)*(x2/2 +C). But if I were to put constants in for the integration factor, I get
    exp(-x)*(C1x2 +C2)
    which is not the same thing.
     
  2. jcsd
  3. Dec 17, 2011 #2
    [tex] y' + py = q [/tex]
    Define [itex]\displaystyle p = P' [/itex]
    [tex] u = e^{\int p\ dx} [/tex]
    [tex] = Pe^C \Rightarrow u = AP [/tex]
    Then [itex]\displaystyle (yu)' = uq [/itex]
    [tex] y = \frac{\int uq\ dx}{u} [/tex]
    [tex] = \frac{\int APq\ dx}{AP} [/tex]
    [tex] = \frac{A}{A}\frac{\int Pq\ dx}{P} [/tex]
    So adding the constant doesn't change anything :)
     
  4. Dec 17, 2011 #3

    nomadreid

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    Gold Member

    Thanks, JHamm.
     
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