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Approximating a 2nd Order ODE

  1. Dec 2, 2007 #1
    1. The problem statement, all variables and given/known data

    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


    3. The attempt at a solution
    The ODE I am trying to approximate involves voltage as a function of time. The examples that I have seen involve position, velocity and acceleration - so there is a relationship (x' = v)

    I am unsure as how to do this in my case.
     
  2. jcsd
  3. Dec 2, 2007 #2
    General Method for Homogeneous Linear Equations:
    av'' + bv'+ cv = 0 (a != 0)
    Write down the characterisitc equation:
    av^2 + bv+ c = 0 (a != 0)

    Determine what type of response it gives by determining the roots. We just want to use b^2 - 4*a*c as it dictates the ODE's transient response.

    1) If both roots are real and distinct (b^2 - 4*a*c > 0) (r1 and r2) then the solution can be written as:
    v(t) = c1*e^(r1*t) + c2*e^(r2*t)

    2) If both roots are real and identical (b^2 - 4*a*c = 0) (r1):
    v(t) = c1*e^(r1*t) + c2*t*e^(r1*t)

    2) If both roots are imaginary (b^2 - 4*a*c < 0) (r1 and r2 conjugate):
    r1 = r2(conj) = x + iy
    x = -b / (2 * a)
    y = sqrt(4*a*c - b^2) / (2 * a)

    v(t) = c1 * e^(x*t)cos(y*t) + c2 * e^(x*t)sin(y*t)
     
  4. Dec 3, 2007 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Dietrick, do you understand that what you did there has no relevance at all to the question?

    Scothoward, with a differential equation of the form d^2v/dt^2 + adv/dt + bv = 0, let u= dv/dt. Then d^2v/dt^2= du/dt and the equation becomes du/dt+ adv/dt+ bv= du/dt+ au+ bv= 0 or du/dt= -au- bv. Of course, your second equation is dv/dt= u.
     
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