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

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

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