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Math Help - Differential equations

  1. #1
    Newbie
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    Oct 2009
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    Differential equations

    Hey
    I am trying to calculate the streamline of u=(ay, -ax, 0)
    I know i need to solve this set of differential equations:
    dx/ds = ay, dy/ds= -ax , dz/ds=0

    I am supposed to get a solution of:
    x = yo sin (as) + xo cos (as)
    y = yo cos (as) - xo sin (as)
    z = zo

    where xo, yo, zo are the intial conditions.
    I don't understand where the sin and cos have come from.
    Thank you for any help
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  2. #2
    Senior Member
    Joined
    Nov 2009
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    Smile

    Quote Originally Posted by capricorn10 View Post
    Hey
    I am trying to calculate the streamline of u=(ay, -ax, 0)
    I know i need to solve this set of differential equations:
    dx/ds = ay, dy/ds= -ax , dz/ds=0

    I am supposed to get a solution of:
    x = yo sin (as) + xo cos (as)
    y = yo cos (as) - xo sin (as)
    z = zo

    where xo, yo, zo are the intial conditions.
    I don't understand where the sin and cos have come from.
    Thank you for any help
    from \frac{1}{a} \frac{dx}{ds} = y and \frac{dy}{ds} = -ax

    we have

    \frac{d^2x}{ds^2} = -a^2x

    which have solution

    y = c_1 \cos (as) + c_2 \sin (as)

    do the rest

    -regards
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  3. #3
    MHF Contributor

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    By the way, to find just the streamlines (without knowing when you are at a particular point on the streamlines), you can eliminate the variable s. Since \frac{dx}{ds}= ay and \frac{dy}{dx}= -ax, \frac{dy}{dx}= \frac{\frac{dy}{ds}}{\frac{dx}{ds}} = \frac{-ax}{ay}= -\frac{x}{y}. That is a separable equation: ydy= -xdx so \frac{1}{2}y^2= -\frac{1}{2}x^2+ C or x^2+ y^2= 2C. Fortunately, \frac{dz}{ds}= 0 so z is a constant. The streamlines are circles, in the plane z= z_0, with center at (0, 0, z_0). You should be able to see that this is exactly what x = yo sin (as) + xo cos (as)
    y = yo cos (as) - xo sin (as)
    z = zo

    considered as parametric equations in the parameter s, give.
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