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Homework Help: PARTIAL FRACTION Help!

  1. Apr 20, 2010 #1
    1. The problem statement, all variables and given/known data
    How to break down:
    [tex]\frac{1}{(s^{2}+1)^{2}}[/tex]
    into partial fractions?


    2. Relevant equations
    -


    3. The attempt at a solution
    I have tried:
    [tex]\frac{1}{(s^{2}+1)^{2}} = \frac{1}{(1+i)^{2}\times(1-i)^{2}} = \frac{A}{(s+i)} + \frac{B}{(s+i)^{2}} + \frac{C}{(s-i)}} + \frac{D}{(s-i)^{2}}[/tex]

    and

    [tex]\frac{1}{(s^{2}+1)^{2}} = \frac{As+B}{(s^{2}+1)^{2}} + \frac{Cs+D}{(s^{2}+1)}[/tex]

    and

    [tex]\frac{1}{(s^{2}+1)^{2}} = \frac{As+B}{(s^{2}+1)} + \frac{Cs+D}{(s^{2}+1)}[/tex]

    but none of them works..
    Please help
     
  2. jcsd
  3. Apr 20, 2010 #2

    tiny-tim

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    Hi v_bachtiar! :wink:

    Why isn't 1/(s2 + 1)2) good enough as it is? :confused:

    But if you do want to break it down further, your first try should have worked …

    what did you get? :smile:
     
  4. Apr 20, 2010 #3
    It is not good enough because I need to perform an inverse Laplace transform on the fraction.
    And at my level, I only use tables and some basic theorems (convolution, shift in s etc.) and 1/(s2 + 1)2) is not on the table :(

    I have attached my working using the first try..
     

    Attached Files:

  5. Apr 20, 2010 #4

    tiny-tim

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    Hi v_bachtiar! :smile:

    What makes you think A B C and D are real? :rolleyes:

    Hint: to simplify it, what are 1/(s - i) ± 1/(s + i) and 1/(s - i)2 ± 1/(s + i)2 ? :wink:
     
  6. Apr 20, 2010 #5
    They are 2s/(s2+1) or 0
    and
    (2s2-2)/(s2+1)2 or 4si/(s2+1)2

    so..

    (As-Ai+Cs+Ci) / (s2+1) + (B(s+i)2+D(s-i)2) / ((s2+1)2) = 1

    and As-Ai+Cs+Ci = 2s , B(s+i)2+D(s-i)2 = 2s2-2

    is this right?
     
  7. Apr 20, 2010 #6
    oh, i mean:

    (As-Ai+Cs+Ci) + (B(s+i)2+D(s-i)2) = 1
     
  8. Apr 21, 2010 #7

    tiny-tim

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    Hi v_bachtiar! :smile:

    (just got up :zzz: …)
    No, 2s/(s2+1) or 2i/(s2+1) :redface:

    ok, rewrite them as

    (2s3+s)/(s2+1)2 or i(2s2+2)/(s2+1)2

    Now can you see how to easily combine them with the others to get 1/(s2+1)2 ? :smile:
     
  9. Apr 21, 2010 #8
    u can do as
    A/s^2+1 + BX/(S^2+1)2
     
  10. Apr 21, 2010 #9
    hi tiny-tim,

    you mean combine (add) them with (2s2-2)/(s2+1)2?

    so 1 = (2s3+s) + (2s2-2)

    then, where do I imply the A, B, C, and D? :confused:

    (thank you for your help so far) :smile:
     
  11. Apr 22, 2010 #10

    tiny-tim

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    How about (2s2+2) and (2s2-2) ? :wink:
     
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