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General solution of a system of equations and partial fractions

  1. Dec 29, 2012 #1
    I've been trying to get out this question for a while now:

    ai) Show that (x,y,z) = (1,1,1) is a solution to the following system of equations:

    x + y + z = 3
    2x + 2y + 2z = 6
    3x + 3y +3z = 9

    aii) Hence find the general solution of the system

    b) Express 2x^2 + 3/(x^2 + 1)^2 in partial fractions

    My attempt:

    Well ai) was simple and i got that part out with barely any effort.

    In aii), i dont even know how to start :( All i know is that the answer is supposed to be:

    (x,y,x) = λ(1,0,-1) + μ(0,1,-1) + (1,1,1,)

    Sorry i cant offer any attempt....i just really dont know where to start....any help at all will be appreciated here.

    With b) i used the matrix method.....but that wasnt the approach they were looking for. I was supposed to use the concept of repeated factors:

    2x^2 + 3/(x^2 + 1)^2 = Ax + B/x^2 + 1 + CX + D/(x^2+1)^2
    (multiply throughout by (x^2 + 1)^2)
    2x^2 +3 = (Ax + B)(x^2 + 1) + Cx + D
    Let x=0
    3 = B + D
    D = 3 - B
    Let x= 1
    5 = (A + B)(2) + C + D
    5 = 2A + 2B + C + D

    Substituting D = 3 - B

    5 = 2A + 2B + C + 3 - B
    2 = 2A + B + C

    Well this is where im stuck.....any help at all would be a life saver. Thank you in advance.
     
  2. jcsd
  3. Dec 29, 2012 #2

    haruspex

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    Strange question. Clearly all three equations are equivalent, so we can ignore the 2nd and 3rd.
    Since x+y+z is linear, having found a solution of x+y+z=3, we can add to it any solution of x+y+z=0 and the result will be a solution of the original equation. So the question is essentially asking you to find all solutions of x+y+z=0.
    Seems you mean (2x2 + 3)/(x2 + 1)2. Please use parentheses properly and subscript/superscript. Makes expressions much more readable.
    The usual procedure from this point is to separate out each power of x into a different equation. That will give you four equations here.
     
  4. Dec 29, 2012 #3

    Ray Vickson

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    In (b), do you mean
    [tex] 2x^2 + \frac{3}{(x^2+1)^2} [/tex] (which is what you wrote), or do you mean
    [tex] \frac{2x^2 + 3}{(x^2 + 1)^2)}? [/tex]
    If you meant the former, then what you wrote is perfectly OK, but if you mean the latter, you must use parentheses, like this: (2x^2 + 3)/(x^2+1)^2. Using ASCII and hence things like x^2 is OK, but you must write clearly. Also, where you write
    Ax + B/x^2 + 1 + CX + D/(x^2+1)^2, you are writing
    [tex] A x + \frac{B}{x^2} + 1 + Cx + \frac{D}{(x^2+1)^2}.[/tex]
    I hope that is not what you really mean, but again, without parentheses, your expressions are impossible to parse (and, frankly, it takes too much of my time, so I won't even try).
     
  5. Dec 29, 2012 #4
    i meant:
    (Ax + B)/(x^2 + 1) + (Cx + D)/((x^2 + 1)^2)

    Sry about that....
     
  6. Dec 29, 2012 #5

    SammyS

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    Expanding the product and collecting terms in the equation

    2x2 +3 = (Ax + B)(x2 + 1) + Cx + D

    gives

    [itex]\displaystyle 2x^2+3=Ax^3+Bx^2+(A+C)x+(B+D)\ .[/itex]

    Now equate coefficients of each power of x.
     
  7. Dec 29, 2012 #6
    Thanks alot guys!!! All help greatly appreciated. :)
     
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