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Math Help - Need help on a system of ODEs?

  1. #1
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    Question Need help on a system of ODEs?

    dx/dt = x(x^3 - 2y^3)

    dy/dt = y(2x^3 - y^3)


    Please help!
    Last edited by hazeleyes; March 5th 2011 at 06:49 AM.
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  2. #2
    Senior Member Sambit's Avatar
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    \frac{dy}{dx} = \frac{y(2x^3-y^3)}{x(x^3-2y^3)}. Notice that both the numerator and denominator are of degree 4,ie homogeneous. Do you know how to solve a homogeneous DE?

    Hint: take y=vx
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  3. #3
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    Quote Originally Posted by Sambit View Post
    \frac{dy}{dx} = \frac{y(2x^3-y^3)}{x(x^3-2y^3)}. Notice that both the numerator and denominator are of degree 4,ie homogeneous. Do you know how to solve a homogeneous DE?

    Hint: take y=vx
    hi, thanks for your hint,
    after i sub y=tx
    i got dy/dx={t(2-t^3)}/{1-2t^3}

    but then what should be doing next? thanks alot
    Last edited by hazeleyes; March 5th 2011 at 04:35 AM.
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  4. #4
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    If y = xv then your ODE becomes

    \dfrac{dv}{dx} = \dfrac{v(1+v^3) }{1-2v^3} \dfrac{1}{x}

    which separate to

    \dfrac{1-2v^3}{v(1+v^3)} dv = \dfrac{dx}{x}

    or

    \left(\dfrac{1}{v} - \dfrac{3v^2}{v^3+1} \right) dv = \dfrac{dx}{x} (this easily integrates!)
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  5. #5
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    what is the 'v' means and what is the reason to sub 'y=vx'?

    it was dy/dx if sub y=vx
    dy= v dx ?

    thanks
    Last edited by hazeleyes; March 5th 2011 at 05:29 AM.
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  6. #6
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    If you have an ODE of the form

    \dfrac{dy}{dx} = F \left(\dfrac{y}{x}\right)

    (called homogeneous) then a substitution y = x v will give another ODE that separates.

    Note y = y(x) \; \text{and}\; v = v(x)
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  7. #7
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    oh I see!! Thank you so much for your clear explanation.
    Last edited by hazeleyes; March 5th 2011 at 06:05 AM.
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  8. #8
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    y=y(x)
    y=vx v=y/x 1/v=x/y

    dy/dx = x(dv/dx)

    x(dv/dx)={ v(2-v^3)}/ (1-2v^3)

    is this form correct? thx
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  9. #9
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    Product rule

    \dfrac{dy}{dx} = x \dfrac{dv}{dx} + v.
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  10. #10
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    Thank you so much for your remind!
    after integration, I got

    v/(v^3 +1) = x + e^c
    v/(v^3 +1) = x + A ,where A is constant
    Attached Thumbnails Attached Thumbnails Need help on a system of ODEs?-shsdh.jpg  
    Last edited by hazeleyes; March 5th 2011 at 06:54 AM.
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