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Math Help - Solving three differential equations

  1. #1
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    Solving three differential equations

    Hi.

    What would be the best way to solve next three differential so that the result it is not dependant from s.
    I know what the x, y, and z are. I need v_x, v_y and v_z.

    <br />
\sqrt{v_x^2+v_y^2+v_z^2} \frac{dv_x}{ds}= -a b \cos(\theta) \frac{dx}{ds}<br />

    <br />
\sqrt{v_x^2+v_y^2+v_z^2} \frac{dv_y}{ds}= -a b \cos(\theta) \frac{dy}{ds}<br />

    <br />
\sqrt{v_x^2+v_y^2+v_z^2} \frac{dv_z}{ds}= -a \frac{dz}{ds} -a b \cos(\theta) \frac{dz}{ds}<br />


    a,\;b are constants.
    Last edited by guest; August 27th 2009 at 07:47 AM.
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  2. #2
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    Quote Originally Posted by guest View Post
    Hi.

    What would be the best way to solve next three differential so that the result it is not dependant from s.
    I know what the x, y, and z are. I need v_x, v_y and v_z.

    <br />
\sqrt{v_x^2+v_y^2+v_z^2} \frac{dv_x}{ds}= -a b \cos(\theta) \frac{dx}{ds}<br />

    <br />
\sqrt{v_x^2+v_y^2+v_z^2} \frac{dv_y}{ds}= -a b \cos(\theta) \frac{dy}{ds}<br />

    <br />
\sqrt{v_x^2+v_y^2+v_z^2} \frac{dv_z}{ds}= -a \frac{dz}{ds} -a b \cos(\theta) \frac{dz}{ds}<br />


    a,\;b are constants.
    What about \theta - is it constant? Also, what is the relation between v_x and x, v_y and y and v_z and z?
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  3. #3
    MHF Contributor chisigma's Avatar
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    Multiplying both terms by ds the first DE becomes...

     \sqrt{v^{2}_{x} + v^{2}_{y} + v^{2}_{z}} \cdot dv_{x} = -a\cdot b \cdot \cos \theta \cdot dx

    ... where the variables are separated. Now we obtain the solution integrating both terms with the aid of the formula...

    \int \sqrt{a^{2} + t^{2}}\cdot  dt = \frac{1}{2} \{a^{2} \cdot \ln (t + \sqrt{a^{2} + t^{2}})+ t \cdot \sqrt{a^{2} + t^{2}}\} + c

    Ther other two differential equations are solvable in the same way...

    Kind regards

    \chi \sigma
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