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Second order homogeneous ODE

  1. Sep 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the values of [itex]α[/itex] for which all the solutions of [itex]y''-(2α-1)y'+α(α-1)y=0[/itex] (a) tend to zero and (b) are ilimited, when [itex]t->∞[/itex].

    2. Relevant equations

    [itex]y''-(2α-1)y'+α(α-1)y=0 => (t)=Ae^{αt}+Be^{(α-1)t}[/itex]

    3. The attempt at a solution

    I found that the general solution to the problem is [itex]y(t)=Ae^{αt}+Be^{(α-1)t}[/itex], which I believe is correct. Then I said that (a) is verified for [itex]t<1/2[/itex] and (b) for [itex]t>=1/2[/itex], but the book's answer is (a) [itex]t<0[/itex] and (b) [itex]t>1[/itex].

    What am I missing?
    Thanks
     
  2. jcsd
  3. Sep 26, 2012 #2

    Simon Bridge

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    As you wrote it, the question asks for values of ##\alpha##, not ##t##.
    So it looks like your (and the book's) answer is for a different question.

    ##y(t)\rightarrow 0## for ##t\rightarrow \infty## when ##\alpha < 0## which makes y(t) a sum of decaying exponentials.
     
  4. Sep 27, 2012 #3
    My mistake, I meant α where I said t. I understand what you said, but then what happens for 0<α<1 ?

    Thanks
     
  5. Sep 27, 2012 #4

    Simon Bridge

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    Well then put that into y(t) ... you have two exponentials added together and ##\alpha## appears in the power.

    What does "ilimited" mean?
     
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