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y(0) y (0) = 0, y(1) + /(1) = 0 y (0) + y (0) = 0, y(1) = 0
11. Consider the general linear homogeneous second order equation
P(x)y+ Q (x) y + R(x) y = 0. (i)
We seek an integrating factor ?(x) such that, upon multiplying Eq. (i) by ?(x), the resulting equation can be written in the form
[^(x) P(x) y\ + ?(x) R(x)y = 0. (ii)
11.1 The Occurrence of Two-Point Boundary Value Problems
(a) By equating coefficients of y, show that j must be a solution of
Pj = ( Q P)j.
(b) Solve Eq. (iii) and thereby show that
Compare this result with that of Problem 27 in Section 3.2.
In each of Problems 12 through 15 use the method of Problem 11 to transform the given equation into the form [p(x)+ q(x)y = 0.
where a2 > 0, c > 0, and k > 0 are constants, is known as the telegraph equation. It arises in the study of an elastic string under tension (see Appendix B of Chapter 10). Equation (i) also occurs in other applications. Assuming that F(x, t) = 0, let u(x, t) = X(x) T(t), separate the variables in Eq. (i), and derive ordinary differential equations for X and T.
17. Consider the boundary value problem
(a) Introduce a new dependent variable u by the relation y = s(x)u. Determine s(x) so that the differential equation for u has no u' term.
(b) Solve the boundary value problem for u and thereby determine the eigenvalues and eigenfunctions of the original problem. Assume that all eigenvalues are real.
(c) Also solve the given problem directly (without introducing u).
18. Consider the boundary value problem
(a) Determine, at least approximately, the real eigenvalues and the corresponding eigenfunctions by proceeding as in Problem 17(a, b).
(b) Also solve the given problem directly (without introducing a new variable).
Hint: In part (a) be sure to pay attention to the boundary conditions as well as the differential equation.
The differential equations in Problems 19 and 20 differ from those in previous problems in that the parameter k multiplies the y term as well as the y term. In each of these problems determine the real eigenvalues and the corresponding eigenfunctions.
(a) Find the determinantal equation satisfied by the positive eigenvalues. Show that there is an infinite sequence of such eigenvalues. Find k: and k2. Then show that kn = [(2n + 1)^/2]2 for large n.
12. y 2x/ + ky = 0,
13. x2y + xy + (x2 v2)y = 0,
14. xy" + (1 x)y + ky = 0,
15. (1 x2)y" xy1 + a2 y = 0,
Hermite equation Bessel equation Laguerre equation Chebyshev equation
16. The equation
utt + cut + ku = a2uxx + F(x, t),
y 2 y + (1 + k) y = 0, y(0) = 0, y(1) = 0.
y + 4y + (4 + 9k) y = 0, y(0) = 0, y( L ) = 0.
19. y" + / + k(y + y) = 0,
y(0) = 0, y(1) = 0
21. Consider the problem
20. x2y k(xy y) = 0,
y(1) = 0, y(2) y (2) = 0
y + ky = 0, 2 y(0) + y(0) = 0, y(1) = 0.
Chapter 11. Boundary Value Problems and Sturm Liouville 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
y" + 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) Ifa < 1, show that 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
y" + Xy = 0, y(0) = 0, y (L) = 0.
Show that if and are eigenfunctions, corresponding to the eigenvalues Xm and Xn,
respectively, with Xm = Xn, then
[ $m (x)$ (x) dx = 0.
Hint: Note that
$"m + Xm $m = 0 < + Xn $n = 0
Multiply the first of these equations by 0n, the second by $m, 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>2 / El; m is the mass per unit length of the rod, E is Youngs modulus, I is the moment of inertia of the cross section about an axis through the centroid perpendicular to the plane of vibration, and m 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 = y = 0, simply supported or hinged end,
y' = y = 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 X: and X2, the two eigenvalues of smallest magnitude. Assume that the eigenvalues are real and positive.