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Math Help - Solving this First-Order ODE

  1. #1
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    Solving this First-Order ODE

    I've been having some real trouble solving this differential equation, mainly because I can't figure out what method to use. The question is as follows:

    Consider the differential equation:

    (x^3/y + 3/x)dx + (3x^3 - x^4/(2y^2) - 1/(2y))dy = 0

    a) Show that the given equation is not exact.
    b) Find constants a and b for which t(x,y) = x^a*y^b is an integrating factor for the given equation.
    c) Use part b) to solve the given equation.

    I think it's easy to show that the ODE is not exact but I'm not sure how to do the rest...

    Thanks a lot,
    AK
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    make your life easier:

    if there's constants m,n\ne0 that verify \dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}=m\dfrac{N}{x}-n\dfrac{M}{y}, the integrating factor is \mu(x,y)=x^my^n.
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  3. #3
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    Ok, so I used that equation and determined that m = -3 and n = 1/2 for the given ODE. How do I use the integrating factor u(x,y)=x^-3*y^(1/2) to solve the ODE. I'm a little bit confused...
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  4. #4
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    pickslides's Avatar
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    Have you tried to re-arrange the original equation to the form y'+f(x)y = g(x) ?
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  5. #5
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    yes, i've tried to reduce it to that form, but I can't successfully move the variables to match that kind of form of ODE.
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