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Homework Help: Partial fraction

  1. Sep 27, 2006 #1
    This is actually part of a larger problem of solving a diff EQ by use of a laplce transform, but it's the partial fraction part that I'm stuck on.

    [tex]\frac{1}{s^2(s^2+w^2)} = \frac{A}{s}+\frac{B}{s^2}+\frac{C*s+D}{s^2+w^2}[/tex]

    Good so far?

    If I then multiply through by the common denominator I get

    EQ.2[tex]1 = A*s(s^2+w^2) + B(s^2+w^2) + s^2(C*s+D)[/tex]

    So then I set s=0

    [tex]1=B*w^2 \longrightarrow B=1/w^2 [/tex]

    Then using the quadratic equation, I find that the roots of s^2+w^2 are +/- jw

    letting s=jw I end up with

    [tex]1 = -jC*w^3 - Dw^2[/tex]

    this is where I get stuck, I'm not sure how to find C and D at this point.

    Note: I just noticed a mistake in EQ.2,
     
    Last edited: Sep 27, 2006
  2. jcsd
  3. Sep 27, 2006 #2
    you can equate the various bits, ie the coefficients of the quadratic linear and constant terms. you cut out a bit of that with s=0 which is a good trick but u need the other way to get all the constants (if only in terms of w)

    ie [tex]{s^2}(C*s + D) = 0[/tex]

    i think
     
  4. Sep 27, 2006 #3
    Thanks for the reply, but I'm not really sure what you are saying.
     
  5. Sep 27, 2006 #4
    You can get two equations for [itex]C[/itex] and [itex]D[/itex] by letting [itex]s=\pm i\omega[/itex]. You can then solve for [itex]C[/itex] and [itex]D[/itex].

    By the way if you want to know a quick way to do this you can say the following:

    [tex]
    \frac{1}{xy} = \frac{1}{x-y}\left\{\frac{1}{y}-\frac{1}{x}\right\}
    [/itex]

    So if I choose [itex]x=s^2+\omega^2[/itex], and [itex]y=s^2[/itex] then your initial fraction can be cast directly into the form on the RHS of your equation.
     
    Last edited: Sep 27, 2006
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