Elementary Differential Equations and Boundary Value Problems 11th Edition solutions

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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 small

Elementary Differential Equations and Boundary Value Problems 11
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 small

Elementary Differential Equations and Boundary Value Problems 11
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 small

Elementary Differential Equations and Boundary Value Problems 11
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 small

Elementary Differential Equations and Boundary Value Problems 11
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 coeffi

Elementary Differential Equations and Boundary Value Problems 11
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)

Elementary Differential Equations and Boundary Value Problems 11
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)

Elementary Differential Equations and Boundary Value Problems 11
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)

Elementary Differential Equations and Boundary Value Problems 11
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)

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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). H

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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 s

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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

Elementary Differential Equations and Boundary Value Problems 11
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 t

Elementary Differential Equations and Boundary Value Problems 11
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