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Math Help - Explaining Sturm-Liouville Problem?

  1. #1
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    Explaining Sturm-Liouville Problem?

    I am having trouble figuring out how to solve sturm-liouville problems. An example of one type of these problems is find the eigen values and eigen vectors of y'' + λy=0 when y'(0)=0, y(1)+y'(1)=0. Another one would be y'' +y' +5λy=0 when y(0)+y'(0)=0 and y(1)+y'(1)=0. Now the way my teacher explained to solve these are you say y=e^(ax) and then sub this into your equation. Then solve for "a". The way the book explains how to do this is that λ=+or - a^2. Can anyone explain to me whats going on and then what to do after finding λ? thanks
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  2. #2
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    Sturm-Liouville is a special type of boundary value problem.

    First consider d2y/dx2 + \lambda*y with  \lambda not equal zero:

    The auxiliary equation r^2 + \lambda = 0 has roots r=\pm \sqrt{-\lambda}.

    Then the solution is of the form y=c_{1}*e^{\sqrt{-\lambda}*x}+c_{2}*e^{-\sqrt{-\lambda}*x}

    Now use the boundary conditions to solve for the constants c_{1} and c_{2}.

    From the boundary conditions y'(0) = 0 and y(1) + y'(1) = 0 you can establish the following system of linear equations:

    let \alpha = \sqrt{-\lambda}

    1. c_{1}\alpha - c_{2}\alpha = 0 ;

    2. c_{1}e^\alpha (1 +\alpha) + c_{2}e^{-\alpha}(1 - \alpha) = 0

    3. Make sure you use the non-trivial solution to c_{1}... I will leave it to you to convince yourself that non-trivial solutions only exist for \lambda > 0.

    The solutions will be in the form of trigonometric functions because the characteristic equation has complex roots.
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