34 An Introduction to Partial Differential Equations with MATLAB©R
FIGURE 1.3
MATLAB graph of the first five eigenfunctions in Example 4, using
the routine BVP4C (highest to lowest, respectively, atxxx=0=0=0... 222 ).
It may seem odd to include such an example, but this problem illustrates
the fact that many eigenvalue problemscannot be solved explicitly.
Further, this type of eigenvalue problem often shows up in applications.
See, for example, Exercises 5e and 5f in Section 4.1.
Example 5Here, we briefly introduce a more general technique for solving
these eigenvalue problems. Suppose we wish to find the positive eigenvalues
of
y′′+λy=0,
y(0) +y′(0) =y(1) = 0.Proceeding as before, we have
y=c 1 coskx+c 2 sinkx, λ=k^2 ,and we must find those values ofkfor which the system
c 1 +c 2 k=0
c 1 cosk+c 2 sink=0.