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    Hey!!!

    Knowing that:
    "The eigenvalue problem  Ly=(py')'+qy, a \leq x \leq b is a Sturm-Liouville problem when it satisfies the boundary conditions:
    p(a)W(u(a),v^*(a))=p(b)W(u(b),v^  *(b)),where W is the wronskian."




    I have to show that the eigenvalue problem y''+λy=0 , with boundary conditions y(0)=0, y'(0)=y'(1) is not a Sturm -Liouville problem.


    This is what I've done so far:


    Let u, v^* solutions of the eigenvalue problem y''+λy=0 , then:
    u(0)=0, u'(0)=u'(1) and v^*(0)=0, v^{*'}(0)=v^{*'}(1).


    W(u(0),v^*(0))=u(0)v^{*'}(0)-u'(0)v^*(0)=0


    W(u(1),v^*(1))=u(1)v^{*'}(1)-u'(1)v^*(1)=u(1) v^{*'}(0)-u'(0)v^*(1)


    How can I continue? How can I show that this is not equal to 0?
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    (Original post by mathmari)
    Hey!!!

    W(u(1),v^*(1))=u(1)v^{*'}(1)-u'(1)v^*(1)=u(1) v^{*'}(0)-u'(0)v^*(1)


    How can I continue? How can I show that this is not equal to 0?
    Could you not find u,v and show it that way? Seems like cheating, but I don't see why you shouldn't do it.
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    (Original post by Smaug123)
    Could you not find u,v and show it that way? Seems like cheating, but I don't see why you shouldn't do it.
    I have found the eigenfunctions of the eigenvalue problem.They are  y_n(x)=\sin(2 n \pi x) .If we suppose that  u and  v^{*} are solutions of the eigenvalue problem,do I have to take two different  n at the eigenfunction to find  u and  v ? I got stuck right now.. :confused:
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    (Original post by mathmari)
    I have found the eigenfunctions of the eigenvalue problem.They are  y_n(x)=\sin(2 n \pi x) .If we suppose that  u and  v^{*} are solutions of the eigenvalue problem,do I have to take two different  n at the eigenfunction to find  u and  v ? I got stuck right now.. :confused:
    Hmm, now I'm confused. Because it's not a Sturm-Liouville problem, we aren't guaranteed to have a complete set of eigenfunctions, so we can't proceed by expressing u and v as a sum of eigenfunctions. I'm stumped, I'm afraid
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    (Original post by Smaug123)
    Hmm, now I'm confused. Because it's not a Sturm-Liouville problem, we aren't guaranteed to have a complete set of eigenfunctions, so we can't proceed by expressing u and v as a sum of eigenfunctions. I'm stumped, I'm afraid
    Oh ok... How did you mean it that I could find u and v? :confused:
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    (Original post by mathmari)
    Oh ok... How did you mean it that I could find u and v? :confused:
    In hindsight, I was assuming that any u and v could be expanded in terms of eigenfunctions. That's true for Sturm-Liouville problems, but not necessarily for non-SL ones.
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    (Original post by Smaug123)
    In hindsight, I was assuming that any u and v could be expanded in terms of eigenfunctions. That's true for Sturm-Liouville problems, but not necessarily for non-SL ones.
    Aha! Ok! Do you have any other idea what I could do?
 
 
 
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