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Existence of solution in Sturm-Liouville DE

  1. Mar 2, 2005 #1
    Hi, I have trouble constructing the proof for the existence of a solution u that vanishes at some point in an open interval (a, b) for the Sturm-Liouville differential equation:

    [tex] \frac{d}{dx} (P(x) \frac{du}{dx}) + Q(x)u = 0 [/tex]

    We can assume that P is continuously differentiable and greater than 0 in the closed interval [a, b], and Q is continuous on [a, b].

    I don't know if it's true that for any second-order ODE, there exists a basis of solutions u1 and u2. Does anyone know? If so, since the Sturm-Liouville DE is of second-order, let u1 and u2 be a basis of solutions, and pick a point c in (a, b). Then we have u(c) = k u1(c) + m u2(c) = 0 by choosing appropriate constants k and m. Not sure if this is right or not.

  2. jcsd
  3. Mar 4, 2005 #2
    We can use the weak variational problem then Lax-Miligram theorem
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