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Math Help - System of differential equations

  1. #1
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    System of differential equations

    Hi guys, suppose we have the following system of differential equations

    X' = -aX + cY
    Y' = aX - (b+c)Y + dZ
    Z' = bY - dZ

    where a,b,c,d > 0 and constant.

    We can show that the system has non-zero constant solutions and that these satisfy

    X/Y = c/a , Y/Z = d/b.

    (This is easy to show)

    But what I now need to do is show that for any initial data, the solution will tend to one of these solutions as t approaches infinity.

    I have tried to find the eigenvalues for the system(using MAPLE) but this gets messy fast.

    If anyone could help it would be much appreciated, cheers.
    Last edited by liedora; August 10th 2012 at 08:15 PM.
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  2. #2
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    Re: System of differential equations

    Hi !
    Note that X'+Y'+Z'=0 hence X+Y+Z=C
    Bringing back Y=C-X-Z into the first and tbe third equation reduces to a system of two equations where the unknown are X and Z, easier to solve. Various methods are allowed (Matricial, or Laplace transform, or substitution)
    For example, sustitution leads to
    X''+(a+b+c+d)X'+(ab+ad+cd)X=Ccd
    Solving this ODE leads to a general solution on the form :
    X(t) = Xoo+C1*exp(k1*t)+C2*exp(k2*t)
    Xoo is a constant term.
    The coeffients C1 and C2 depend on the initial condition, but it doesn't matter.
    k1 and k2 are rather simple functions of a, b, c, d which can be easily expressed.
    It is then possible to show that k1 and k2 are negative as far as a, b, c, d are positive. So, when t tends to infinity, the exponential terms tends to 0. Only the constant term remains, which is the limit as t approches the infinity.
    Last edited by JJacquelin; August 11th 2012 at 12:42 AM.
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