This problem illustrates that the eigenvalue parameter sometimes appears in the boundary

In each of Problems 1 through 6, state whether the given boundary value problem is homogeneous or nonhomogeneous. y__ + 4y = 0, y(1) = 0, y(1) = 0

In each of Problems 1 through 6, state whether the given boundary value problem is homogeneous or nonhomogeneous. ((1 + x2) y_)_ + 4y = 0, y(0) = 0, y(1) = 1

In each of Problems 1 through 6, state whether the given boundary value problem is homogeneous or nonhomogeneous. y__ + 4y = sin x, y(0) = 0, y(1) = 0

In each of Problems 1 through 6, state whether the given boundary value problem is homogeneous or nonhomogeneous. y__ + x2 y = y, y_(0) y(0) = 0, y_(1) + y(1) = 0

In each of Problems 1 through 6, state whether the given boundary value problem is homogeneous or nonhomogeneous. ((1 + x2) y_)_ = y + 1, y(1) = 0, y(1) = 0

In each of Problems 1 through 6, state whether the given boundary value problem is homogeneous or nonhomogeneous. y__ = (1 + x2) y, y(0) = 0, y_(1) + 3y(1) = 0

In each of Problems 7 through 10, assume that all eigenvalues are real. a. Determine the form of the eigenfunctions and the determinantal equation satisfied by the nonzero eigenvalues. b. Determine whether = 0 is an eigenvalue. c. Find approximate values for 1 and 2, the nonzero eigenvalues of smallest absolute value. d. Estimate n for large values of n y__ + y = 0, y(0) = 0, y( ) + y_( ) = 0

In each of Problems 7 through 10, assume that all eigenvalues are real. a. Determine the form of the eigenfunctions and the determinantal equation satisfied by the nonzero eigenvalues. b. Determine whether = 0 is an eigenvalue. c. Find approximate values for 1 and 2, the nonzero eigenvalues of smallest absolute value. d. Estimate n for large values of n y__ + y = 0, y_(0) = 0, y(1) + y_(1) = 0

In each of Problems 7 through 10, assume that all eigenvalues are real. a. Determine the form of the eigenfunctions and the determinantal equation satisfied by the nonzero eigenvalues. b. Determine whether = 0 is an eigenvalue. c. Find approximate values for 1 and 2, the nonzero eigenvalues of smallest absolute value. d. Estimate n for large values of n y__ + y = 0, y(0) y_(0) = 0, y(1) + y_(1) = 0

In each of Problems 7 through 10, assume that all eigenvalues are real. a. Determine the form of the eigenfunctions and the determinantal equation satisfied by the nonzero eigenvalues. b. Determine whether = 0 is an eigenvalue. c. Find approximate values for 1 and 2, the nonzero eigenvalues of smallest absolute value. d. Estimate n for large values of n y__ y = 0, y(0) + y_(0) = 0, y(1) = 0

Consider the general linear homogeneous second-order equation P( x) y__ + Q( x) y_ + R( x) y = 0. (30) We seek an integrating factor ( x) such that, upon multiplying equation (30) by ( x), we can write the resulting equation in the form (( x) P( x) y_)_ + ( x) R( x) y = 0. (31) a. By equating coefficients of y_ in equations (30) and (31), show that must be a solution of P _ = ( Q P_). (32) b. Solve equation (32) and thereby show that ( x) = 1 P( x) exp __ x x0 Q(s) P(s) ds . (33) Compare this result with that of Problem 31 in Section 3.2.

Compare this result with that of Problem 31 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) y) + q( x) y = 0. y__ 2xy_ + y = 0 (Hermite equation)

Compare this result with that of Problem 31 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) y) + q( x) y = 0. x2 y__ + xy_ + ( x2 2) y = 0 (Bessel equation)

Compare this result with that of Problem 31 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) y) + q( x) y = 0. xy__ + (1 x) y_ + y = 0 (Laguerre equation)

Compare this result with that of Problem 31 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) y) + q( x) y = 0. (1 x2) y__ xy_ + 2 y = 0 (Chebyshev equation)

The equation utt + cut + ku = a2uxx + F( x, t), (34) 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 (34) also occurs in other applications. Consider equation (34) with F( x, t) = 0. Let u( x, t) = X( x)T (t), separate variables, and derive ordinary differential equations for X and T .

Consider the boundary value problem y__ 2y_ + (1 + ) y = 0, y(0) = 0, y(1) = 0. a. Introduce a new dependent variable u by the relation y = s( x)u. Find the equivalent boundary value problem in terms of 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. Solve the given problem directly (without introducing u).

Consider the boundary value problem y__ + 4y_ + (4 + 9) y = 0, y(0) = 0, y_( L) = 0. a. Determine, at least approximately, the real eigenvalues and the corresponding eigenfunctions by proceeding as in Problem 17a and 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 multiplies the y_ term as well as the y term. In each of these problems, determine the real eigenvalues and the corresponding eigenfunctions.

The differential equations in Problems 19 and 20 differ from those in previous problems in that the parameter multiplies the y term as well as the y term. In each of these problems, determine the real eigenvalues and the corresponding eigenfunctions. y__ + y_ + ( y_ + y) = 0, y_(0) = 0, y(1) = 0

The differential equations in Problems 19 and 20 differ from those in previous problems in that the parameter multiplies the y term as well as the y term. In each of these problems, determine the real eigenvalues and the corresponding eigenfunctions. x2 y__ ( xy_ y) = 0, y(1) = 0, y(2) y_(2) = 0

Consider the problem y__ + y = 0, 2y(0) + y_(0) = 0, y(1) = 0. a. Find the determinantal equation satisfied by the positive eigenvalues. b. Show that there is an infinite sequence of such eigenvalues. c. Find 1 and 2. Then show that n = ((2n + 1)/2)2 for large n. d. Find the determinantal equation satisfied by the negative eigenvalues. e. Show that there is exactly one negative eigenvalue and find its value.

Consider the problem y__ + y = 0, y(0) + y_(0) = 0, y(1) = 0, where is a given constant. a. Show that for all values of there is an infinite sequence of positive eigenvalues. b. If < 1, show that all (real) eigenvalues are positive. Show that the smallest eigenvalue approaches zero as approaches 1 from below. c. Show that = 0 is an eigenvalue only if = 1. d. If > 1, show that there is exactly one negative eigenvalue and that this eigenvalue decreases as increases.

Consider the problem y__ + y = 0, y(0) = 0, y_( L) = 0. Show that if m and n are eigenfunctions corresponding to the eigenvalues m and n, respectively, with m _= n, then _ L 0 m( x)n( x)dx = 0. Hint: Note that __ m + mm = 0, __ n + nn = 0. Multiply the first of these equations by n, the second by m, and integrate from 0 to L, using integration by parts. Finally, subtract one equation from the other.

In this problem we consider a higher-order eigenvalue problem. The analysis of transverse vibrations of a uniform elastic bar is based on the differential equation y(4) y = 0, where y is the transverse displacement and = m2/E I; 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 is the frequency of vibration. Thus, for a bar whose material and geometrical 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 1 and 2, the two eigenvalues of smallest magnitude. Assume that the eigenvalues are real and positive. a. y(0) = y__(0) = 0, y( L) = y__( L) = 0 b. y(0) = y__(0) = 0, y( L) = y_( L) = 0 c. y(0) = y_(0) = 0, y__( L) = y___( L) = 0 (cantilevered bar)

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 and cross-sectional area A. It can be shown that the axial displacement u( x, t) satisfies the partial differential equation E uxx = utt ; 0 < x < L, t > 0, (35) where E is Youngs modulus and 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. (36) 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 Newtons 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. (37) a. Assume that u( x, t) = X( x)T (t), and show that X( x) and T (t) satisfy the differential equations X__ + X = 0, (38) T __ + E T = 0. (39) b. Show that the boundary conditions are X(0) = 0, X_( L) LX( L) = 0, (40) where = m AL is a dimensionless parameter that gives the ratio of the end mass to the mass of the bar. Hint: Use the differential equation for T (t) in simplifying the boundary condition at x = L. c. Determine the form of the eigenfunctions and the equation satisfied by the real eigenvalues of equations (38) and (40). d. Find the first two eigenvalues 1 and 2 if = 0.5.