Sturm-Liouville Problems and Eigenfunction Expansions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One framework for every vibrating string

Mathematics · Differential Equations · ages 21-22
Name ______________________   Date ____________

  1. The problem y'' + λ y = 0 with y(0) = 0 and y(π) = 0 has eigenvalues n^2 for n = 1, 2, 3 and so on. What is the smallest eigenvalue?

    Answer: ______________

  2. The functions 1 and x are eigenfunctions of a Legendre boundary value problem with weight 1 on [-1, 1]. Work out the integral from -1 to 1 of x dx.

    Answer: ______________

  3. The equation y'' + (2/x) y' + w y = 0 is put into self-adjoint form. What is the weight function?

    • x squared
    • 2x
    • 1

  4. The problem y'' + w y = 0 with y(0) = 0 and y(pi) = 0 has eigenvalues 1, 4, 9, and so on. What is the smallest eigenvalue?

    • 0
    • 9
    • 1

  5. The function h(x) = 3 sin(4x) minus 2 sin(x) is expanded in modes sin(nx). What is the coefficient of sin(4x)?

    • Minus 2
    • 3
    • 1

  6. The function f(x) = 2 sin(x) + 5 sin(2x) - sin(3x) is expanded in the eigenfunctions sin(nx) on [0, π]. What is the coefficient of sin(3x)?

    • -1
    • 1
    • 3
    • -3

  7. The equation y'' + (2/x) y' + λ y = 0 is put into Sturm-Liouville form. What is the weight function?

    • x^2
    • x
    • 2x
    • 1

  8. The function f(x) = 2 sin(x) + 5 sin(2x) minus sin(3x) is expanded in modes sin(nx). What is the coefficient of sin(3x)?

    • 5
    • Minus 1
    • 2

  9. A classmate claims any two solutions of the same boundary value problem are automatically orthogonal. True or false?

    Circle one:   True   False

  10. A student expands f but forgets the weight in the coefficient integrals. When does the error stay hidden?

    • Whenever the weight equals 1, as with the plain sine modes
    • Whenever the interval is very long
    • The weight never matters in any problem
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Answer key

For grown-ups. Fold this page away before handing over the rest.

One framework for every vibrating string W1-mt_oqadAPaSsW-s1

  1. 1 · The eigenvalues are 1, 4, 9 and so on, and the smallest of these is 1.
  2. 0 · The antiderivative of x is x^2/2, which is 1/2 at both ends, so the definite integral is 1/2 - 1/2 = 0.
  3. x squared · Multiplying by x squared collapses the first terms into (x^2 y')'.
  4. 1 · The list starts at one, and nothing smaller appears.
  5. 3 · Only the 3 sin(4x) term survives projection onto sin(4x).
  6. -1 · The expansion is already written as a sum of eigenfunctions, so the coefficient of sin(3x) is the number multiplying it, which is -1.
  7. x^2 · Multiplying by x^2 turns the equation into (x^2 y')' + λ x^2 y = 0, so the weight function is x^2.
  8. Minus 1 · The expansion is already written out, so read off the multiplier of sin(3x).
  9. False · Orthogonality needs distinct eigenvalues; two modes sharing one eigenvalue need not be orthogonal.
  10. Whenever the weight equals 1, as with the plain sine modes · With unit weight the forgotten factor changes nothing, but any other weight breaks the coefficients.
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