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A challenging ODE question: find an integrating factor

  1. Oct 29, 2011 #1
    This is the ODE: y' + siny + xcosy + x = 0.
    The problem is: Find an integrating factor for the ODE above.

    You can see my solution to the ODE here: https://www.physicsforums.com/showthread.php?t=543662. from my solution it seems that [tex]e^x(sec^2(y/2))[/tex] must be an integrating factor. but I fail to show that it's an integrating factor.
    I guess it must be truly challenging to find an integrating factor for it :(
     
  2. jcsd
  3. Oct 29, 2011 #2

    lurflurf

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    I think that is an integrating factor, recall the general test
    u is an integrating factor of N y'+M if

    My-Nx=(ux/u)N-(uy/u)M
    where subscripts are partial derivatives and for this equation we have
    M=sin(y) + x cos(y) + x
    N=1
    u=exsec2(y/2)=2ex/(1+cos(y))
    you will need to use some simple trigonometric identities
     
    Last edited: Oct 29, 2011
  4. Oct 29, 2011 #3
    What he says doesn't make sense to me to be honest. I've solved the ODE, and I'm 99% sure that exsec2(y/2) is an integrating factor. he tells me that I should multiply the integrating factor and show that it turns the ODE into an exact differential because his problem asks me to find an integrating factor for it, not solving the ODE.
     
    Last edited: Oct 29, 2011
  5. Oct 29, 2011 #4

    lurflurf

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    Yes if My-Nx=(ux/u)N-(uy/u)M
    then the differential is exact this follows from requiring
    (uM)y-(uN)x=0
    then it is possible to choose a P such that
    N=Py/u
    M=Px/u
    hence
    (P)'/u=N y'+M
    as desired
     
    Last edited: Oct 29, 2011
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