Page 328 - Schaum's Outline of Differential Equations

CHAP. 32] SECOND-ORDER BOUNDARY-VALUE PROBLEMS 311

Property 32.3. For each eigenvalue of a Sturm-Liouville problem, there exists one and only one linearly
independent eigenfunction.
[By Property 32.3 there corresponds to each eigenvalue A, n a unique eigenfunction with lead coefficient
unity; we denote this eigenfunction by e n(x).]

Property 32.4. The set of eigenfunctions {e^x), e 2(x), ...} of a Sturm-Liouville problem satisfies the relation

for n i= m, where w(x) is given in Eq. (32.6).

Solved Problems

32.1. Solve /' + 2/ - 3y = 0; y(0) = 0, /(I) = 0.
This is a homogeneous boundary-value problem of the form (32.3), with P(x) = 2, Q(x) = -3, a : = 1, fi 1 = 0,
3
x
a 2= 0, /? 2= 1, a = 0, and b=l. The general solution to the differential equation is y = cf * + c 2e Applying the
boundary conditions, we find that c 1 = c 2 = 0; hence, the solution is y = 0.
x
The same result follows from Theorem 32.1. Two linearly independent solutions arey\(x) = e^ andy 2 (x) = e*',
hence, the determinant (32.5) becomes

Since this determinant is not zero, the only solution is the trivial solution y(x) = 0.

32.2. Solve /' = 0; y(-l) = 0, y(l) - 2/(l) = 0.
This is a homogeneous boundary-value problem of form (32.3), where P(x) = Q(x) = 0, KI = 1, Pi = 0, O 2 = 1,
/? 2 = -2, a = -1, and b=l. The general solution to the differential equation is y = Cj+ c 2x. Applying the boundary
conditions, we obtain the equations c 1 - c 2 = 0 and c 1 - c 2 = 0, which have the solution c 1 = c 2, c 2 arbitrary. Thus, the
solution to the boundary-value problem is y = c 2(l +x), c 2 arbitrary. As a different solution is obtained for each
value of c 2, the problem has infinitely many nontrivial solutions.
The existence of nontrivial solutions is also immediate from Theorem 32.1. Here y\(x) = 1, y 2(x) = x, and deter-
minant (32.5) becomes

32.3. Solve /' + 2/ -3y = 9x; y(0) = 1, /(I) = 2.
This is a nonhomogeneous boundary-value problem of forms (32.1) and (32.2) where $(x) =x, Yi= L and
%=2. Since the associated homogeneous problem has only the trivial solution (Problem 32.1), it follows from
Theorem 32.2 that the given problem has a unique solution. Solving the differential equation by the method of
Chapter 11, we obtain

Applying the boundary conditions, we find

whence

Finally,