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Mathematics LibreTexts

11.1E: Eigenvalue Problems for y'' + λy = 0 (Exercises)

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    18284
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    Exercises [exer:11.1.19}- [exer:11.1.22} ask you to verify that the eigenfunctions of Problems 1-4 are orthogonal on \([0,L]\). However, this also follows from a general theorem that we’ll prove in Chapter 13.

    [exer:11.1.1] Prove that \(\lambda=0\) is an eigenvalue of Problem 5 with associated eigenfunction \(y_0=1\), and that any other eigenvalues must be positive.

    [exer:11.1.2] \(y''+\lambda y=0,\quad y(0)=0,\quad y(\pi)=0\)

    [exer:11.1.3] \(y''+\lambda y=0,\quad y'(0)=0,\quad y'(\pi)=0\)

    [exer:11.1.4] \(y''+\lambda y=0\),\(y(0)=0\),\(y'(\pi)=0\)

    [exer:11.1.5] \(y''+\lambda y=0\),\(y'(0)=0\),\(y(\pi)=0\)

    [exer:11.1.6] \(y''+\lambda y=0,\quad y(-\pi)= y(\pi), \quad y'(-\pi)=y'(\pi)\)

    [exer:11.1.7] \(y''+\lambda y=0,\quad y'(0)=0,\quad y'(1)=0\)

    [exer:11.1.8] \(y''+\lambda y=0\),\(y'(0)=0\),\(y(1)=0\)

    [exer:11.1.9] \(y''+\lambda y=0,\quad y(0)=0,\quad y(1)=0\)

    [exer:11.1.10] \(y''+\lambda y=0,\quad y(-1)= y(1), \quad y'(-1)=y'(1)\)

    [exer:11.1.11] \(y''+\lambda y=0\),\(y(0)=0\),\(y'(1)=0\)

    [exer:11.1.12] \(y''+\lambda y=0,\quad y(-2)= y(2), \quad y'(-2)=y'(2)\)

    [exer:11.1.13] \(y''+\lambda y=0,\quad y(0)=0,\quad y(2)=0\)

    [exer:11.1.14] \(y''+\lambda y=0\),\(y'(0)=0\),\(y(3)=0\)

    [exer:11.1.15] \(y''+\lambda y=0\),\(y(0)=0\),\(y'(1/2)=0\)

    [exer:11.1.16] \(y''+\lambda y=0,\quad y'(0)=0,\quad y'(5)=0\)

    [exer:11.1.17] Prove Theorem [thmtype:11.1.3}.

    [exer:11.1.18] Prove Theorem [thmtype:11.1.5}.

    [exer:11.1.19] Verify that the eigenfunctions

    \[\sin{\pi x\over L}, \, \sin{2\pi x\over L},\dots, \, \sin{n\pi x\over L},\dots\]

    of Problem 1 are orthogonal on \([0,L]\).

    [exer:11.1.20] Verify that the eigenfunctions

    \[1,\, \cos{\pi x\over L}, \, \cos{2\pi x\over L},\dots, \, \cos{n\pi x\over L},\dots\]

    of Problem 2 are orthogonal on \([0,L]\).

    [exer:11.1.21] Verify that the eigenfunctions

    \[\sin{\pi x\over 2L}, \, \sin{3\pi x\over 2L},\dots, \, \sin{(2n-1)\pi x\over 2L},\dots\]

    of Problem 3 are orthogonal on \([0,L]\).

    [exer:11.1.22] Verify that the eigenfunctions

    \[\cos{\pi x\over 2L}, \, \cos{3\pi x\over 2L},\dots, \, \cos{(2n-1)\pi x\over 2L},\dots\]

    of Problem 4 are orthogonal on \([0,L]\).

    [exer:11.1.23] \(y''+\lambda y=0,\quad y(0)=0,\quad \int_0^Ly(x)\,dx=0\)

    [exer:11.1.24] \(y''+\lambda y=0,\quad y'(0)=0,\quad \int_0^Ly(x)\,dx=0\)

    [exer:11.1.25] \(y''+\lambda y=0,\quad y(L)=0,\quad \int_0^Ly(x)\,dx=0\)

    [exer:11.1.26] \(y''+\lambda y=0,\quad y'(L)=0,\quad \int_0^Ly(x)\,dx=0\)