Section 6.3 Eigenvalues

Objective:

1. Eigenvalues and functions of boundary value problems

In previous section, we learn that we must solve the boundary value problem: [latex]X''+\lambda X=0[/latex], [latex]X(0)=0[/latex] and [latex]X(L)=0[/latex] or [latex]X'(0)=0[/latex] and [latex]X'(L)=0[/latex]. At here, we are solving more general type of questions: what value [latex]\lambda[/latex] such that the boundary value problem [latex]X''+\lambda X=0[/latex], with one of four types boundary conditions:  (1) [latex]X(0)=0[/latex] and [latex]X(L)=0[/latex];  (2) [latex]X'(0)=0[/latex] and [latex]X(L)=0[/latex];  (3) [latex]X(0)=0[/latex] and [latex]X'(L)=0[/latex];  (4) [latex]X'(0)=0[/latex] and [latex]X'(L)=0[/latex]; would have a non-trivial solution. Notice [latex]X=0[/latex] is always a solution. When the [latex]\lambda[/latex] does give us non-trivial solution, we call this [latex]\lambda[/latex] an eigenvalue and the non-trivial solution is called an eigenfunction. Our first step is to exclude a particular class of [latex]\lambda[/latex], when [latex]\lambda \lt 0[/latex]. 

Example 1: Prove that when [latex]\lambda \lt 0,[/latex] it is not an eigenvalue of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X(0)=0[/latex], and [latex]X(L)=0[/latex]. 

Exercise 1: Prove that when [latex]\lambda \lt 0,[/latex] it is not an eigenvalue of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X'(0)=0[/latex], and [latex]X'(L)=0[/latex]. 

Example 2:  Find all eigenvalues and their corresponding eigenfunctions of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X'(0)=0[/latex], and [latex]X'(L)=0[/latex]. 

Exercise 2: Find all eigenvalues and their corresponding eigenfunctions of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X(0)=0[/latex], and [latex]X(L)=0[/latex]. 

Example 3: Find all eigenvalues and their corresponding eigenfunctions of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X(0)=0[/latex], and [latex]X'(L)=0[/latex]. 

Exercise 3: Find all eigenvalues and their corresponding eigenfunctions of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X'(0)=0[/latex], and [latex]X(L)=0[/latex]. 

Group work:

1. Find all eigenvalues and their corresponding eigenfunctions of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X'(0)=0[/latex], and [latex]X'(4)=0[/latex]. 

2. Find all eigenvalues and their corresponding eigenfunctions of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X(0)=0[/latex], and [latex]X(9)=0[/latex]. 

3. Find all eigenvalues and their corresponding eigenfunctions of the boundary problem [latex]X''+\lambda X=0[/latex], [latex]X'(0)=0[/latex], and [latex]X(16)=0[/latex].