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Partial Fraction Decomposition—Multiple Variables

  1. Dec 4, 2014 #1

    END

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    What's the best approach to solving the partial-fraction decomposition of the following expression?

    $$\frac{1}{(a-y)(b-y)}$$

    The expression is not of the following forms:

    upload_2014-12-4_18-39-1.png

    But I know the solution is

    $$= \frac{1}{(a-b)(y-a)}-\frac{1}{(a-b)(y-b)}$$

     

    Attached Files:

  2. jcsd
  3. Dec 4, 2014 #2

    ShayanJ

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    Gold Member

    I don't use such tables. Any time I want to do a partial fraction decomposition, I just write (e.g.) [itex] \frac{1}{(a-y)(b-y)}=\frac{A}{a-y}+\frac{B}{b-y} [/itex] and then determine A and B.
    Anyway, if you multiply the factors you'll see that its in fact in the form of the third entry in the table!
     
  4. Dec 5, 2014 #3

    Mark44

    Staff: Mentor

    What the table is saying is the for each distinct (i.e., not repeated) factor (ax + b) in the denominator, you'll have a term ##\frac{A}{ax + b}## in the decomposition. So ##\frac{1}{(a - y)(b - y)}## results in ##\frac{A}{a - y} + \frac{B}{b - y}##.

    Equate the two expressions and solve for A and B, which is more or less what Shyan said.
     
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