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Homework Help: Integrating factor x^n*y^m

  1. Jan 31, 2006 #1
    Hey everyone,
    I need to find an integrating factor of the form x^n*y^m, to solve a differential equation i have... however i do not know the process to solve for an integration of this form.. .any help??
    Thanks
    Steph
     
  2. jcsd
  3. Jan 31, 2006 #2
    can you give an example problem that your working on?
     
  4. Jan 31, 2006 #3
    the problem is ( 12 + 5xy )dx + (6 (x/y)+ 3x^2)dy =0
    and it says, find an integrating factor of the form (x^n) * (y^m), and solve the equation...
    thanks sweetie
    steph
     
  5. Feb 1, 2006 #4

    HallsofIvy

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    Science Advisor

    RULE 1: Mathematics problems are not solved by staring at a problem until you remember the answer! They are solved by plugging things in and doing the algebra.
    So TRY!!

    If you multiply the equation by [itex]x^ny^m[/itex] you get
    [itex](12x^ny^m+ 5x^{n+1}y^{m+1})dx+ (6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)dy= 0[/itex]

    In order for that to be an exact equation, you must have
    [itex](12x^ny^m+ 5x^{n+1}y^{m+1})_y= (6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)_x[/itex]

    Do the derivatives and see what m and n must be for those to be equal!

    [itex](12x^ny^m+ 5x^{n+1}y^{m+1})_y= 12mx^ny^{m-1}+ 5(m+1)x^{n+1}y^m[/itex]
    [itex](6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)_x= 6(n+1)x^ny^{m-1}+3(n+2)x^{n+1}y^m[/itex]
    Coefficients of the same powers must be equal. That gives two equations for m and n.
     
    Last edited by a moderator: Feb 1, 2006
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