1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Re(eigenvalue) inequality problem

  1. Sep 28, 2005 #1

    N2

    User Avatar

    Hello,
    if a diff.eqn. has the characteristic equation [itex]\lambda^2 + (3-K) \lambda + 1 = 0[/itex]
    the eigenvalues solves to [itex]\lambda=-3/2 + K/2 \pm 1/2 \sqrt{5-6K+K^2}[/itex]. No problem there. But when is the diff.eqn. asymp. stable, meaning [itex]\Re(\lambda)<0[/itex] ?

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

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

    Thanks
     
  2. jcsd
  3. Sep 29, 2005 #2

    CarlB

    User Avatar
    Science Advisor
    Homework Helper

    My inclination would be to first consider the case when [tex]5-6K+K^2[/tex] is positive, and then consider the case when it is negative. If you separate them out, it shouldn't be too hard.

    Carl
     
  4. Sep 30, 2005 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Solving K2- 6K+ 5> 0 tells us that there will be complex roots for K between 1 and 5 and real roots for K<= 1, >= 5.

    I there are complex roots, the real part is just -3/2+ K/2. That is 0 for K= 3, negative for K< 3, positive for K> 3. The solution will be stable for 1< K< 3, unstable for 3< K< 5.

    For K<= 1 or K>= 5, we need to look at all of [itex]-3/2 + K/2 \pm 1/2 \sqrt{5-6K+K^2}[/itex]

    The best way to determine where that is positive or negative is to set it equal to 0 and solve the resulting quadratic equation. Those will separate "< 0" from "> 0". Choose one value of K in each resulting interval to see whether this is positive or negative.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Re(eigenvalue) inequality problem
  1. Inequality problem (Replies: 5)

  2. Eigenvalue problem (Replies: 2)

Loading...