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ODE System Help

  1. Jun 21, 2011 #1
    Hello all,
    I don't have much experience with ODEs.

    I have a simple system, which I believe is first order linear, similar to the following:

    dA/dt = 2A + 3B - C

    dB/dt = A + 2B - C

    dC/dt = -2A + 5B - 2C

    Now I would like to include the constraint that A + B + C = 1. How do I do this mathematically?
     
  2. jcsd
  3. Jun 21, 2011 #2

    tiny-tim

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    welcome to pf!

    hello mykat! welcome to pf! :smile:
    dA/dt + dB/dt + dC/dt = 0 :wink:
     
  4. Jun 21, 2011 #3

    hunt_mat

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    Or, just write C=1-A-B and insert it in the first two equations to obtain:
    dA/dt=3A+2B-1
    dB/dt=2A+3B-1
     
  5. Jun 21, 2011 #4
    Thank you for the replies. I appreciate the input, I had thought to use a similar method but I wasn't sure if it was applicable.

    Unfortunately, I have 7 equations and 7 variables, and as I am working with matlab, I need to have them each in the form similar to dA/dt = 3A + 4B...

    Is there a more general analytical approach, rather than algebraically working out all of the equations by hand?
     
  6. Jun 21, 2011 #5

    hunt_mat

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    Exponentials of matrices, so if you write in your example [itex]\mathbf{X}=(A,B,C)^{T}[/itex], the, you can write your equations in the form:
    [tex]
    \frac{d\mathbf{X}}{dt}=\mathbf{J}\mathbf{X}
    [/tex]
    From here you can diagonalise your J and then solve it very easily. Can can be automatically done in matlab.
     
    Last edited: Jun 21, 2011
  7. Jun 21, 2011 #6
    By T do you mean transpose? If so, I initially had the matrices in that form. After that I wanted to add the A + B + C = 1 condition, without working out and modifying each line by hand. Is there a way to do this?

    Sorry if I've completely misunderstood you.
     
  8. Jun 21, 2011 #7

    hunt_mat

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    T does mean transpose. As for the A+B+C=1 condition, it's only 7 equations, or do you mean to increase it later?
     
  9. Jun 21, 2011 #8
    Only 7 equations.
     
  10. Jun 21, 2011 #9

    hunt_mat

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    Then it's not that bad then, once you've done that little hardship then you can apply my method as a quick way of solving the system.
     
  11. Jun 22, 2011 #10
    As it turns out, the constraint was completely unnecessary. The time I wasted on this problem yesterday reflects my poor understanding of differential equations.

    I am actually working with a Markov model, where the initial conditions dictate that state 1 has probability = 1 and all others are zero. Based on the nature of differential equations, probability is conserved when the system is modeled correctly.

    Initially I had made a small error in the model, which gave me strange results and the false idea that I had to include a constraint. This was a great learning experience. I only wish the class I took on diff eq 2 years ago were this useful to me.

    Thanks for the help anyway.
     
  12. Jun 22, 2011 #11

    hunt_mat

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    So you can solve the system without any problems now?
     
  13. Jun 22, 2011 #12
    Sure. Solving it was never the issue, it was including the unity condition, which as it happens is not necessary.
     
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