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

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