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Math Help - [SOLVED] Help with "Exact Differential Equations"

  1. #1
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    [SOLVED] Help with "Exact Differential Equations"

    I need to find integrating factor, i will solve the equation once the integrating factor is found.

    Equation:
    (10x^2+2xy+6y^2)dx+(3x^2+4xy+5y^2)dy=0

    Where the equation is in the form of: (M)dx+(N)dy=0

    \dfrac{d}{dy}[M]=2x+12y

    \dfrac{d}{dx}[N]=6x+4y

    \dfrac{d}{dy}[M] \not = \dfrac{d}{dx}[N] Therefore Differential equation is NOT exact...yet, I still need to find the integrating factor.

    I need to find an "integrating factor" using this special case:
    u(x,y)=ax+by
    \mu=\mu(ax+by) if an only if \dfrac{M_{y}-N_{x}}{aN-bM}=R(u)=\dfrac{\mu'(u)}{\mu(u)}

    \dfrac{M_{y}-N_{x}}{aN-bM} should simpify down to 1 (i think, professor didnt say for this special case ) by making a and b appropriote constant values. Here is what I get:

    \dfrac{-4x+8y}{(3a-10b)x^2+(4a-2b)xy+(5a-6b)y^2}

    I cannot make a and b values that will simplify this fraction down to "1" unless they are 0, which does not work.

    I need to find the "integrating factor" but messed up or am doing it wrong, in either case any help would be appreciated. Thanks!
    Last edited by snaes; February 1st 2010 at 08:58 PM.
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  2. #2
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    Hey, isn't it homogeneous? Why not solve it via y=vx?
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  3. #3
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    That would make this problem a lot easier, except I dont know how to do a problem similar to this, where I need to find an integrating factor in the form ax+by, to make the equation exact.
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