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Math Help - I could use a second opinion on the completeness of my solution...

  1. #1
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    I could use a second opinion on the completeness of my solution...

    I could use a second opinion on my answer to this question:


    Determine the values of a, if any, for which all solutions to the equation:

    y'' - (2a - 1)y' + a(a - 1)y = 0

    ...tend to zero as t approaches infinity.



    The general solution comes out to this:


    y = \frac{C_1e^{at}}{e^t} + C_2e^{at}


    So, my answer would be this:


    0 < a < 1

    if

    C_2 = 0



    a = 0

    if

    C_2 \neq 0


    Is my answer correct, or did I miss something?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: I could use a second opinion on the completeness of my solution...

    It is not correct. The condition is \lim_{t\to +\infty}x(t)=0 for all solutions x(t) . Then, you have to find the values of a such that \lim_{t\to +\infty}(C_1e^{(a-1)t}+C_2e^{at})=0 for all C_1,C_2
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  3. #3
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    Re: I could use a second opinion on the completeness of my solution...

    I'm not quite sure how to proceed, then.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: I could use a second opinion on the completeness of my solution...

    Quote Originally Posted by Lancet View Post
    I'm not quite sure how to proceed, then.

    Use the following: \lim_{t\to +\infty}e^{\beta t}=0\Leftrightarrow \beta <0 .
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  5. #5
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    Re: I could use a second opinion on the completeness of my solution...

    So is it correct that the general answer would be:

    a < 0
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: I could use a second opinion on the completeness of my solution...

    Quote Originally Posted by Lancet View Post
    So is it correct that the general answer would be: a < 0
    Right.
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    Re: I could use a second opinion on the completeness of my solution...

    Quote Originally Posted by FernandoRevilla View Post
    Right.
    Thank you very much for your help!
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    Re: I could use a second opinion on the completeness of my solution...

    Quick followup:

    If a second question for this same problem asked for the values of a for which all nonzero solutions become unbounded as t approaches infinity, would that just mean a > 0 ?
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  9. #9
    MHF Contributor FernandoRevilla's Avatar
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    Re: I could use a second opinion on the completeness of my solution...

    Quote Originally Posted by Lancet View Post
    If a second question for this same problem asked for the values of a for which all nonzero solutions become unbounded as t approaches infinity, would that just mean a > 0 ?
    It is not right. Choose for instance C_1=1,C_2=0, a=1/2 .
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    Re: I could use a second opinion on the completeness of my solution...

    Quote Originally Posted by FernandoRevilla View Post
    It is not right. Choose for instance C_1=1,C_2=0, a=1/2 .
    Ah yes, thank you.

    That being the case, a > 1 would fill that requirement, would it not?
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  11. #11
    MHF Contributor FernandoRevilla's Avatar
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    Re: I could use a second opinion on the completeness of my solution...

    Quote Originally Posted by Lancet View Post
    That being the case, a > 1 would fill that requirement, would it not?

    Yes, if (C_1,C_2)\neq (0,0) and a>1 then, x(t)=e^{(a-1)t}(C_1+C_2e^t)\to (+\infty\;\textrm{or}\; -\infty) as t\to +\infty .
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  12. #12
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    Re: I could use a second opinion on the completeness of my solution...

    Thank you again for all your help!
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