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(b) Show that if q (x) > 0 and if b1 / b2 and — a1 /a2 are nonnegative, then the eigenvalue k is nonnegative.
(c) Under the conditions of part (b) show that the eigenvalue k is strictly positive unless q(x) = 0 for each x in 0 < x < 1 and also a1 = b1 = 0.
22. Derive Eq. (8) using the inner product (9) and assuming that u and v are complex-valued functions.
Hint: Consider the quantity L [u] v dx, split u and v into real and imaginary parts, and
proceed as in the text.
23. In this problem we indicate a proof that the eigenfunctions of the Sturm-Liouville problem (1), (2) are real.
(a) Let k be an eigenvalue and ô a corresponding eigenfunction. Let ô(x) = U(x) + iV(x), and show that U and V are also eigenfunctions corresponding to k.
(b) Using Theorem 11.2.3, or the result of Problem 20, show that U and V are linearly dependent.
(c) Show that ô must be real, apart from an arbitrary multiplicative constant that may be complex.
24. Consider the problem
xV = k(xy - y), y(1) = 0, y(2) = 0.
Note that k appears as a coefficient of y as well as of y itself. It is possible to extend the definition of self-adjointness to this type of problem, and to show that this particular problem is not self-adjoint. Show that the problem has eigenvalues, but that none of them is real. This illustrates that in general nonself-adjoint problems may have eigenvalues that are not real.
Buckling of an Elastic Column. In an investigation of the buckling of a uniform elastic column of length L by an axial load P (Figure 11.2.1 a) one is led to the differential equation
yv + ky" = 0, 0 < x < L. (i)
The parameter k is equal to P/ EI, where E is Young’s modulus and I is the moment of inertia of the cross section about an axis through the centroid perpendicular to the xy-plane. The boundary conditions at x = 0 and x = L depend on how the ends of the column are supported. Typical boundary conditions are
y = y = 0, clamped end; y = y' = 0, simply supported (hinged) end.
The bar shown in Figure 11.2.1 a is simply supported at x = 0 and clamped at x = L. It is desired to determine the eigenvalues and eigenfunctions of Eq. (i) subject to suitable boundary conditions. In particular, the smallest eigenvalue k1 gives the load at which the column buckles, or can assume a curved equilibrium position, as shown in Figure 11.2.1 b. The corresponding eigenfunction describes the configuration of the buckled column. Note that the differential equation (i) does not fall within the theory discussed in this section. It is possible to show, however, that in each of the cases given here all the eigenvalues are real and positive. Problems 25 and 26 deal with column buckling problems.
11.3 Nonhomogeneous Boundary Value Problems
Ó Ó ? " ---1 ,
³ \ x Lx
x = 0 x = L
FIGURE 11.2.1 (a) A column under compression. (b) Shape of the buckled column.
25. For each of the following boundary conditions find the smallest eigenvalue (the buckling load) of yv + Xy" = 0, and also find the corresponding eigenfunction (the shape of the buckled column).
(a) y(0) = y"(0) = 0, y(L) = y"(L) = 0
(b) y(0) = /(0) = 0, y(L) = Ó(L) = 0
(c) y(0) = /(0) = 0, y(L) = Ó (L) = 0
26. In some buckling problems the eigenvalue parameter appears in the boundary conditions
as well as in the differential equation. One such case occurs when one end of the column
is clamped and the other end is free. In this case the differential equation ylv + ky/' = 0 must be solved subject to the boundary conditions
y (0) = 0, Ó (0) = 0, Ó'(L) = 0, Ó" (L) + êÓ( L) = 0.
Find the smallest eigenvalue and the corresponding eigenfunction.
11.3 Nonhomogeneous Boundary Value Problems
In this section we discuss how to solve nonhomogeneous boundary value problems, both for ordinary and partial differential equations. Most of our attention is directed toward problems in which the differential equation alone is nonhomogeneous, while the boundary conditions are homogeneous. We assume that the solution can be expanded in a series of eigenfunctions of a related homogeneous problem, and then determine the coefficients in this series so that the nonhomogeneous problem is satisfied. We first describe this method as it applies to boundary value problems for second order linear ordinary differential equations. Later we illustrate its use for partial differential equations by solving a heat conduction problem in a bar with variable material properties and in the presence of source terms.