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Mathematics LibreTexts

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

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    18284
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    Q11.1.1

    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.

    Q11.1.2

    In Exercises 11.1.2-11.1.16 solve the eigenvalue problem.

    2. \(y''+\lambda y=0,\quad y(0)=0,\quad y(\pi)=0\)

    3. \(y''+\lambda y=0,\quad y'(0)=0,\quad y'(\pi)=0\)

    4. \(y''+\lambda y=0,\quad y(0)=0,\quad y'(\pi)=0\)

    5. \(y''+\lambda y=0,\quad y'(0)=0,\quad y(\pi)=0\)

    6. \(y''+\lambda y=0,\quad y(-\pi)= y(\pi), \quad y'(-\pi)=y'(\pi)\)

    7. \(y''+\lambda y=0,\quad y'(0)=0,\quad y'(1)=0\)

    8. \(y''+\lambda y=0,\quad y'(0)=0,\quad y(1)=0\)

    9. \(y''+\lambda y=0,\quad y(0)=0,\quad y(1)=0\)

    10. \(y''+\lambda y=0,\quad y(-1)= y(1), \quad y'(-1)=y'(1)\)

    11. \(y''+\lambda y=0,\quad y(0)=0,\quad y'(1)=0\)

    12. \(y''+\lambda y=0,\quad y(-2)= y(2), \quad y'(-2)=y'(2)\)

    13. \(y''+\lambda y=0,\quad y(0)=0,\quad y(2)=0\)

    14. \(y''+\lambda y=0,\quad y'(0)=0,\quad y(3)=0\)

    15. \(y''+\lambda y=0,\quad y(0)=0,\quad y'(1/2)=0\)

    16. \(y''+\lambda y=0,\quad y'(0)=0,\quad y'(5)=0\)

    Q11.1.3

    17. Prove Theorem 11.1.3.

    18. Prove Theorem 11.1.5.

    19. Verify that the eigenfunctions

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

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

    20. Verify that the eigenfunctions

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

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

    21. Verify that the eigenfunctions

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

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

    22. Verify that the eigenfunctions

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

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

    Q11.1.4

    In Exercises 11.1.23-11.1.26 solve the eigenvalue problem.

    23. \(y''+\lambda y=0,\quad y(0)=0,\quad \int_0^Ly(x)\,dx=0\)

    24. \(y''+\lambda y=0,\quad y'(0)=0,\quad \int_0^Ly(x)\,dx=0\)

    25. \(y''+\lambda y=0,\quad y(L)=0,\quad \int_0^Ly(x)\,dx=0\)

    26. \(y''+\lambda y=0,\quad y'(L)=0,\quad \int_0^Ly(x)\,dx=0\)