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Thread: Approximating a 2nd Order ODE

  1. #1
    Newbie
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    Oct 2007
    Posts
    13

    Approximating a 2nd Order ODE

    Hey,

    I am trying to approximate the solution to a second order ODE using the 4th order Runge-Kutta.

    I was told that in order to do this, I have to write the second order ODE and a pair of 1st order ODEs.

    Given that my differential equation is

    d^2v/dt^2 + adv/dt + bv = 0, where a and b are constant coefficients, I am a little lost on how to do this.

    Any advice on how to approach this?

    Thanks
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  2. #2
    Senior Member
    Joined
    Dec 2007
    From
    Anchorage, AK
    Posts
    276

    Converting to a system

    To convert a single second-order ODE to a system of two couples first-order ODEs, we create a new function equal to the derivative of our unknown function: we define $\displaystyle w = \dot v$. Then $\displaystyle \dot w = \ddot v = -a\dot v - bv = -aw-bv$. So then our system is $\displaystyle \dot v = w, \dot w = -aw-bv$, or in matrix notation,

    $\displaystyle \begin{pmatrix}
    \dot v \\
    \dot w
    \end{pmatrix} = \begin{pmatrix}
    0 & 1 \\
    -b & -a
    \end{pmatrix}\begin{pmatrix}
    v \\
    w
    \end{pmatrix}$


    From here, you then perform the fourth-order Runge-Kutta on this system.


    --Kevin C.
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