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: $X^{″}+\lambda X=0$, $X(0)=0$ and $X(L)=0$ or $X^{\prime }(0)=0$ and $X^{\prime }(L)=0$. At here, we are solving more general type of questions: what value $\lambda$ such that the boundary value problem $X^{″}+\lambda X=0$, with one of four types boundary conditions:  (1) $X(0)=0$ and $X(L)=0$;  (2) $X^{\prime }(0)=0$ and $X(L)=0$;  (3) $X(0)=0$ and $X^{\prime }(L)=0$;  (4) $X^{\prime }(0)=0$ and $X^{\prime }(L)=0$; would have a non-trivial solution. Notice $X=0$ is always a solution. When the $\lambda$ does give us non-trivial solution, we call this $\lambda$ an eigenvalue and the non-trivial solution is called an eigenfunction. Our first step is to exclude a particular class of $\lambda$, when $\lambda <0$. 

Example 1: Prove that when $\lambda <0,$ it is not an eigenvalue of the boundary problem $X^{″}+\lambda X=0$, $X(0)=0$, and $X(L)=0$. 

Exercise 1: Prove that when $\lambda <0,$ it is not an eigenvalue of the boundary problem $X^{″}+\lambda X=0$, $X^{\prime }(0)=0$, and $X^{\prime }(L)=0$. 

Example 2:  Find all eigenvalues and their corresponding eigenfunctions of the boundary problem $X^{″}+\lambda X=0$, $X^{\prime }(0)=0$, and $X^{\prime }(L)=0$. 

Exercise 2: Find all eigenvalues and their corresponding eigenfunctions of the boundary problem $X^{″}+\lambda X=0$, $X(0)=0$, and $X(L)=0$. 

Example 3: Find all eigenvalues and their corresponding eigenfunctions of the boundary problem $X^{″}+\lambda X=0$, $X(0)=0$, and $X^{\prime }(L)=0$. 

Exercise 3: Find all eigenvalues and their corresponding eigenfunctions of the boundary problem $X^{″}+\lambda X=0$, $X^{\prime }(0)=0$, and $X(L)=0$. 

Group work:

1. Find all eigenvalues and their corresponding eigenfunctions of the boundary problem $X^{″}+\lambda X=0$, $X^{\prime }(0)=0$, and $X^{\prime }(4)=0$. 

2. Find all eigenvalues and their corresponding eigenfunctions of the boundary problem $X^{″}+\lambda X=0$, $X(0)=0$, and $X(9)=0$. 

3. Find all eigenvalues and their corresponding eigenfunctions of the boundary problem $X^{″}+\lambda X=0$, $X^{\prime }(0)=0$, and $X(16)=0$.