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Math Help - Help with Solving for Constants in Partial Fraction Integration

  1. #1
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    Help with Solving for Constants in Partial Fraction Integration

    In the middle of a very large partial fractions integration problem, I have reached the stage where the constants must be solved for. I have double checked these equations and they are correct, but given the scale of them I'm having difficulty solving for all the variables.

    A+3B+3D+3F=0
    -2A+5B+3C-D+3E-4F+3G=0
    5A+15B+11C+13D-E+6F-4G+3H=0
    -14A+13B+37B-11D-13E+19F+6G-7H+3I=0
    -A-47B+87C+5D-11E-60F-19G+10H-7I=0
    -8A-123B+127C+37D+5E+50F-60G-22H+10I=0
    34A-184B+131C+32D+37E+72F+50G-48H-22I=0
    24A-126B+78C+20D+32E+120F+72G+120H-48I=0
    24A-60B+30C+20E+120G+120I=1
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  2. #2
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    Do you have the original problem? If indeed you have a 9x9 to solve who's your instructor the Marquis de Sade ?

    Obviously the way to go if you did have to solve the system is to use

    a computer and matrix methods.

    I've included the answer done in Mathcad
    Attached Thumbnails Attached Thumbnails Help with Solving for Constants in Partial Fraction Integration-matrix.jpg  
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  3. #3
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    Quote Originally Posted by PD1337 View Post
    In the middle of a very large partial fractions integration problem, I have reached the stage where the constants must be solved for. I have double checked these equations and they are correct, but given the scale of them I'm having difficulty solving for all the variables.

    A+3B+3D+3F=0
    -2A+5B+3C-D+3E-4F+3G=0
    5A+15B+11C+13D-E+6F-4G+3H=0
    -14A+13B+37B-11D-13E+19F+6G-7H+3I=0
    -A-47B+87C+5D-11E-60F-19G+10H-7I=0
    -8A-123B+127C+37D+5E+50F-60G-22H+10I=0
    34A-184B+131C+32D+37E+72F+50G-48H-22I=0
    24A-126B+78C+20D+32E+120F+72G+120H-48I=0
    24A-60B+30C+20E+120G+120I=1
    Yes, please do post the original problem.
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  4. #4
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    The original problem was the integration of 1/[(3x+5)(x-2)^2(x^2+6)(x^2+x+1)^2]

    It was meant to combine all 4 of the partial fraction integration cases into one problem.

    Could you enlighten me as to how to decipher the matrix? Is the top of the row to the far right a, the second from the top b, etc?
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  5. #5
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    Quote Originally Posted by PD1337 View Post
    The original problem was the integration of 1/[(3x+5)(x-2)^2(x^2+6)(x^2+x+1)^2]

    It was meant to combine all 4 of the partial fraction integration cases into one problem.

    Could you enlighten me as to how to decipher the matrix? Is the top of the row to the far right a, the second from the top b, etc?
    One thing you can do (although long) is split your problem into subproblems. For example, one would try the following

     <br />
\frac{1}{ (x-1)(x+1) } = \frac{A}{x-1} + \frac{B}{x+1}.<br />

    However, note that

    (x+1) - (x-1) = 2 \;\; \text{so}\;\; \frac{1}{2}(x+1) - \frac{1}{2}(x-1) = 1 and

     <br />
\frac{1}{2} \cdot \frac{(x+1) - (x-1)}{(x-1)(x+1) } = \frac{1}{2} \left( \frac{1}{x-1} - \frac{1}{x+1}\right).<br />

    This could work for you. For example,

    5(x^2 + 6) - (x-2)^2 - 4(x^2+x+1) = 22

    so replacing your numerator by \frac{1}{22} \cdot \big(5(x^2 + 6) - (x-2)^2 - 4(x^2+x+1)\big) and expanding would replace your problem with three different problems but with one term reduced. Then repeat. Again, long but it could work.
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  6. #6
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    Possibility

    If you're familiar with matrix algebra, you can solve them that way.
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