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Math Help - Re(eigenvalue) inequality problem

  1. #1
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    Re(eigenvalue) inequality problem

    Hej,
    if a diff.eqn. has the characteristic equation L^2 + (3-k)L +1 = 0
    the eigenvalues solves to -3/2 + K/2 +/- 1/2*sqrt(5-6K+K^2). No problem there. But when is the diff.eqn. asymp. stable, meaning Re(L)<0 ?

    I can only get this far
    Re[ -3/2 + K/2 +/- 1/2*sqrt(5-6K+K^2) ] < 0
    -3/2 + 1/2 Re[ K +/- sqrt(5-6K+K^2) ] < 0

    How can i find the values for K, where this inequality is true?

    Thanks
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  2. #2
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    Firstly distinguish the cases when the eigenvalues are real and complex. When the discriminant k^2-6k+5 < 0, the eigenvalues are a complex conjugate pair with real part (k-3)/2. Now k^2-6k+5 < 0 precisely when 2 < k < 3. So in this case we have (k-3)/2 < 0 and there is stability. Otherwise the discriminant is positive and there are two real roots: we want to know whether they are noth negative. The product of the roots is 1, so they have the same sign. The sum of the roots is k-3 and this is negative if k < 3. But the case k > 2 has already been dealt with -- that leads to imaginary roots. So there is stability with two negative real roots if k <= 2. The remaining case is k>=3 which means two positive real roots and hence instability.
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