Partial Differential Equations with MATLAB

Introduction 33

FIGURE 1.2

MATLAB graph of the intersection of the functionsyyy===−−−kkk and
yyy=tan=tan=tankkkfork>k>k> 000.

Therefore, the eigenvalues are thoseλn>0 satisfying−

√

λn=tan

√

λn,

with associated eigenfunctions

yn=sin

√

λnx.

We have also solved this same problem using the MATLAB routine BVP4C.

The first five eigenvalues are given in Table 1.1, and the first five eigenfunctions

(normalizedby requiring thaty′(0) = 1) are plotted in Figure 1.3. Note that

the solutionsdoseem to satisfy the conditiony′(1) =−y(1).

nλn

1. 4.116


2. 24.142


3. 63.664


4. 122.897


5. 201.863

TABLE 1.1

First five eigenvalues of the problem in Example 4, computed using

the MATLAB routine BVP4C.