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I Convert an ODE to matrix form

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  1. Nov 11, 2017 #1

    SeM

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    Hi, I have the following ODE:

    aY'' + bY' + c = 0

    I would like to convert it to a matrix, so to evaluate its eigenvalues and eigenvectors. I have done so for phase.plane system before, however there were two ODEs there. In this case, there is only one, so how does this look like in a matrix form given that this is non homogenous? Is it as such:

    \binom{a \ \ 1}{-b \ \ 1}\binom{c}{0}}

    Or how is it correct to convert ONE ODE to matrix form, and from there study it?
     
  2. jcsd
  3. Nov 11, 2017 #2
    you can write a second order ode as a system of two first order ode's by introducing new variables:
    ## y_1 = Y ##
    ## y_2 = \frac{dY}{dx} ##

    such that the original ode becomes:
    ## a\frac{dy_2}{dx} + by_1 +c = 0##
    with the auxiliary ode:
    ## y_2 = \frac{dy_1}{dx} ##
     
  4. Nov 12, 2017 #3

    SeM

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    Thanks!
     
  5. Nov 14, 2017 #4

    SeM

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    Is there any chance of getting the constant c in the matrix form at all, or should a matrix form rather give the homogenous form in order to solve the eigenvalues?
     
  6. Nov 14, 2017 #5
    Well, do you think the eigenvalues will change by the constant c?
     
  7. Nov 15, 2017 #6

    SeM

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    No, they won't indeed! Thanks!
     
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