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Ó + 4y + (4 + 9k) y = 0, y(0) = 0, / (L) = 0.
19. Ó + y + k(y + y) = 0,
y(0) = 0, y(1) = 0
21. Consider the problem
20. x2Ó — k(xy — y) = 0,
y(1) = 0, y(2) — y (2) = 0
Ó + ky = 0, 2 y(0) + y (0) = 0, y(1) = 0.
(a) Find the determinantal equation satisfied by the positive eigenvalues. Show that there is an infinite sequence of such eigenvalues. Find kj and k2. Then show that kn = [(2n + 1)æ/2]2 for large n.
Chapter 11. Boundary Value Problems and SturmLiouville Theory
(b) Find the determinantal equation satisfied by the negative eigenvalues. Show that there is exactly one negative eigenvalue and find its value.
22. Consider the problem
Ó + Xy = 0, ay(0) + y (0) = 0, y(1) = 0,
where a is a given constant.
(a) Show that for all values of a there is an infinite sequence of positive eigenvalues.
(b) If a < 1,showthat all (real) eigenvalues are positive. Show that the smallest eigenvalue approaches zero as a approaches 1 from below.
(c) Show that X = 0 is an eigenvalue only if a = 1.
(d) If a > 1, show that there is exactly one negative eigenvalue and that this eigenvalue decreases as a increases.
23. Consider the problem
Ó + Xy = 0, y(0) = 0, y(L) = 0.
Show that if ôò and ôï are eigenfunctions, corresponding to the eigenvalues Xm and Xn, respectively, with Xm = Xn, then
[ Ôò (Õ)Ôï (x) dx = 0.
Hint: Note that
Ô'ò + XmÔ m = 0, ô'ï + Xn Ôï = 0
Multiply the first of these equations by Ôï, the second by Ôø, and integrate from 0 to L, using integration by parts. Finally, subtract one equation from the other.
24. In this problem we consider a higher order eigenvalue problem. In the study of transverse vibrations of a uniform elastic bar one is led to the differential equation
yv — Xy = 0,
where y is the transverse displacement and X = ma? / EI; m is the mass per unit length of the rod, E is Young’s modulus, I is the moment of inertia of the cross section about an axis through the centroid perpendicular to the plane of vibration, and a is the frequency of vibration. Thus for a bar whose material and geometric properties are given, the eigenvalues determine the natural frequencies of vibration. Boundary conditions at each end are usually one of the following types:
y = y = 0, clamped end,
y = Ó = 0, simply supported or hinged end,
Ó = Ó" = 0, free end.
For each of the following three cases find the form of the eigenfunctions and the equation satisfied by the eigenvalues of this fourth order boundary value problem. Determine X1 and X2, the two eigenvalues of smallest magnitude. Assume that the eigenvalues are real and positive.
(a) y(0) = Ó(0) = 0, y(L) = Ó'(L) = 0
(b) y(0) = /(0) = 0, y(L) = y(L) = 0
(c) y(0) = /(0) = 0, y'(L) = y"(L) = 0 (cantilevered bar)
25. This problem illustrates that the eigenvalue parameter sometimes appears in the boundary
conditions as well as in the differential equation. Consider the longitudinal vibrations of a uniform straight elastic bar of length L. It can be shown that the axial displacement u(x, t) satisfies the partial differential equation
( E /P)uxx = utt;
0 < x < L, t > 0,
11.2 Sturm-Liouville Boundary Value Problems
where E is Young’s modulus and p is the mass per unit volume. If the end x = 0 is fixed, then the boundary condition there is
u(0, t) = 0, t > 0. (ii)
Suppose that the end x = L is rigidly attached to a mass m but is otherwise unrestrained.
We can obtain the boundary condition here by writing Newton’s law for the mass. From the theory of elasticity it can be shown that the force exerted by the bar on the mass is given by — EAux(L, t). Hence the boundary condition is
EAux(L, t) + mutt(L, t) = 0, t > 0. (iii)
(a) Assume that u(x, t) = X(x) T(t), and showthat X(x) and T(t) satisfy the differential
X" + k X = 0, (iv)
T" + k( E/p)T = 0. (v)