11.1E: Eigenvalue Problems for y'' + λy = 0 (Exercises)
- Page ID
- 18284
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. \(y''+\lambda y=0,\quad y'(L)=0,\quad \int_0^Ly(x)\,dx=0\)