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Question on Variation of Parameters

  1. Mar 19, 2010 #1
    I have a question on the integration part of the Variation of Parameters. Given .[tex]y''+P(x)y'+Q(x)y=f(x)[/tex]

    The associate homogeneous solution .[tex] y_c=c_1y_1 + c_2y_2[/tex].

    The particular solution .[tex] y_p=u_1y_1 + c_2y_2[/tex].

    [tex]u'_1 = -\frac{W_1}{W} = -\frac{y_2f(x)}{W} [/tex]

    This is where I have question. Some books use indefinite integral with the integration constant equal 0.

    [tex]u_1= -\int \frac{y_2f(x)}{W}dx[/tex]

    But other books gave:

    [tex]u_1= -\int_{x_0}^x \frac{y_2f(s)}{W}ds[/tex]

    Where [tex]x_0[/tex] is any number in I.

    None of the books explain this. Can anyone explain to me about this?

    Last edited: Mar 19, 2010
  2. jcsd
  3. Mar 19, 2010 #2


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    If yp = u1y1 + u2y2 is a particular solution you found, then compare this to:

    Yp = (u1+A)y1 + (u2+B)y2

    =u1y1 + u2y2 + Ay1+ By2

    When you plug the second expression into the equation the last two terms give 0 because they satisfy the homogeneous equation. So the constants don't give anything extra.
  4. Mar 19, 2010 #3
    Thanks for the respond.

    So you mean even using definite integral, substituding in x0 only produce a constant as in your example of A and B. These will be absorbed into the associate homogeneous part ( into c1 and c2).
  5. Mar 23, 2010 #4
    Does it matter? We only want a particular solution.
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