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Math Help - Convert to Exact diff eq

  1. #1
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    Convert to Exact diff eq

    I start with the equation:
    (-xysinx +2ycosx)dx +(2xcosx)dy with the integrating factor given u(x,y)=xy
    after multiplying by u I get:
    (-x^2 * y^2 * sinx +2xy^2 * cosx) +(2x^2 * ycosx)y'=0

    then taking partial derivatives for M and N
    Msuby= -2x^2*y*sinx +4xycosx
    Nsubx= -4xysinx

    I get Msuby does not equal Nsubx

    Do I have to find another integrating factor for this equation or is there some calculus/algebra rule I'm not seeing where Msuby does equal Nsubx?
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  2. #2
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    Re: Convert to Exact diff eq

    (-xysinx +2ycosx)dx +(2xcosx)dy=0
    There is no need for integrating factor, since it is a separable ODE :
    2dy/y = ((xsinx -2cosx)/xcosx)dx
    2 ln(y) = -ln(cos(x)) -2 ln(x) +c
    y = C/(xcos(x))
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  3. #3
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    Re: Convert to Exact diff eq

    Thanks I guess that works, but I figured out I forgot to perform the product rule when taking my partial derivatives which makes Msuby=Nsubx.
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