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Math Help - Finding the Integrating Factor

  1. #1
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    Finding the Integrating Factor

    Question: For what values of  m and  n will  u = x^ny^m be an integrating factor for the differential equation

     (-3y-2x)dx + (5x+4x^2y^{-1})dy = 0

    The exact differential equation which results from multiplying by this integrating factor has solution  F(x,y) = C where  F(x,y) =___________________________.

    Attempt at solving the question:

    First I check to see whether either differential is exact (equal to each other), of course they are not.

    The continue to solve and reach this point:

     u'(y)(-3y -2x) + u(y)(-3-2x) = u(y)(5+8xy^{-1})
     (-3y -2x)u'(y) + (-3-2x-5-8x^{-1}) = 0
     (-3y -2x)u'(y) + (-8-2x-8xy^{-1}) = 0

    I should be able to factor something out, but i don't seem to find any, so am I doing the question wrong?
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  2. #2
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    Krizalid's Avatar
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    did you already solve this? i saw the "solved" tag above but i don't actually see that you solved your problem.
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  3. #3
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    Actually, i did that because no one was responding. Anyways, I did figure out my integrating factor to be  x^{-1}, n = -1

    I get to a point where after i multiplied both parts of the equation by the integrating factor and then i took the integral of the part of the equation and i get my F(x,y) which is  -1/7x^4y^7+ (1/3)x^9y^8 + x^6/6 + K(y) . After wards i took the partial derivative with respect to y and got  -x^4y^6 + (8/3)x^9y^7 + k'(y) . I equate both equations and i get  K'(y) = 0 , which if i integrated k'(y), i would get C since. But my ans i entered was incorect.

     F(x,y) = (-1/7)x^4y^7 +(-1/3)x^9y^8 (-1/6)x^6 = C
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  4. #4
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    Krizalid's Avatar
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    you can actually find the integrating factor of the form \mu(x,y)=x^my^n, the equation has it when for m,n\ne0 constants we have \dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}=m\dfrac{N}{x}-n\dfrac{M}{y}.

    it's a necessary and sufficient condition, so if it doesn't hold, don't waste your time.

    -----

    on the other hand always compute \dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x} and see its form, it would give ya an idea if you can find an integrating factor depending exclusively of x or y.

    here's a hint, multiply both sides by y and apply the method to find the integrating factor you were asked to find.
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