Fundamentals of Differential Equations [With CDROM] - 7th Edition
Fundamentals of Differential Equations [With CDROM] - 7th Edition
7th Edition
ISBN: 9780321410481
Author: Saff, Edward B., Snider, Arthur David, Nagle, R. Kent
Publisher: Addison Wesley
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Chapter 8.2, Problem 32E

In Problems 2 9 − 3 4 , determine the Taylor series about the point x 0 for the given functions and values of x 0 .

f ( x ) = ln ( 1 + x ) , x 0 = 0

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A factorization A = PDP 1 is not unique. For A= 7 2 -4 1 1 1 5 0 2 1 one factorization is P = D= and P-1 30 = Use this information with D₁ = to find a matrix P₁ such that - -1 -2 0 3 1 - - 1 05 A-P,D,P P1 (Type an integer or simplified fraction for each matrix element.)
Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 30 -1 - 1 0 -1 400 0 0 1 A= 3 4 3 0 1 3 040 3 1 3 0 0 4 1 0 0 003 -1 0 -1 Select the correct choice below and fill in the answer boxes to complete your choice. (Use a comma to separate vectors as needed.) A basis for the corresponding eigenspace is { A. There is one distinct eigenvalue, λ = B. In ascending order, the two distinct eigenvalues are λ₁ ... = and 2 = Bases for the corresponding eigenspaces are { and ( ), respectively. C. In ascending order, the three distinct eigenvalues are λ₁ = = 12/2 = and 3 = Bases for the corresponding eigenspaces are {}, }, and { respectively.

Chapter 8 Solutions

Fundamentals of Differential Equations [With CDROM] - 7th Edition

Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 18, determine the first three nonzero...Ch. 8.1 - In Problems 1-8, determine the first three nonzero...Ch. 8.1 - In Problems 1-8, determine the first three nonzero...Ch. 8.1 - In Problems 1-8, determine the first three nonzero...Ch. 8.1 - a. Construct the Taylor polynomial p3(x) of degree...Ch. 8.1 - a. Construct the Taylor polynomial p3(x) of degree...
Ch. 8.1 - Prob. 11ECh. 8.1 - Prob. 12ECh. 8.1 - Duffings Equation. In the study of a nonlinear...Ch. 8.1 - Soft versus Hard Springs. For Duffings equation...Ch. 8.1 - Prob. 15ECh. 8.1 - van der Pol Equation. In the study of the vacuum...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - In Problems 1-6, determine the convergence set of...Ch. 8.2 - Prob. 7ECh. 8.2 - Determine the convergence set of the given power...Ch. 8.2 - In Problems 9 and 10, find the power series...Ch. 8.2 - In Problems 9 and 10, find the power series...Ch. 8.2 - In Problems 11-14, find the first three nonzero...Ch. 8.2 - In Problems 11-14, find the first three nonzero...Ch. 8.2 - Prob. 13ECh. 8.2 - In Problems 11-14, find the first three nonzero...Ch. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prob. 17ECh. 8.2 - In Problems 17-20, find a power series expansion...Ch. 8.2 - Prob. 19ECh. 8.2 - In Problems 17-20, find a power series expansion...Ch. 8.2 - Prob. 21ECh. 8.2 - In Problems 21 and 22, find a power series...Ch. 8.2 - Prob. 23ECh. 8.2 - In Problems 23-26, express the given power series...Ch. 8.2 - Prob. 25ECh. 8.2 - In Problems 23-26, express the given power series...Ch. 8.2 - Prob. 27ECh. 8.2 - Show that...Ch. 8.2 - In Problems 29-34, determine the Taylor series...Ch. 8.2 - In Problems 2934, determine the Taylor series...Ch. 8.2 - Prob. 31ECh. 8.2 - In Problems 2934, determine the Taylor series...Ch. 8.2 - Prob. 33ECh. 8.2 - In Problems 2934, determine the Taylor series...Ch. 8.2 - Prob. 35ECh. 8.2 - Let f(x) and g(x) be analytic at x0. Determine...Ch. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - Prob. 4ECh. 8.3 - In Problem 110, determine all the singular points...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 110, determine all the singular points...Ch. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - Prob. 13ECh. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - Prob. 16ECh. 8.3 - In Problems 1118, find at least the first four...Ch. 8.3 - Prob. 18ECh. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 1924, find a power series expansion...Ch. 8.3 - In Problems 19-24, find a power series expansion...Ch. 8.3 - In Problems 25-28, find at least the first four...Ch. 8.3 - Prob. 26ECh. 8.3 - In Problems 25-28, find at least the first four...Ch. 8.3 - In Problems 25-28, find at least the first four...Ch. 8.3 - Prob. 29ECh. 8.3 - Prob. 30ECh. 8.3 - In Problems 29-31, use the first few terms of the...Ch. 8.3 - Prob. 32ECh. 8.3 - Use the ratio test to show that the radius of...Ch. 8.3 - Prob. 34ECh. 8.3 - Prob. 35ECh. 8.3 - Variable Spring Constant. As a spring is heated,...Ch. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - Prob. 5ECh. 8.4 - In Problems 16, find a minimum value for the...Ch. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - Prob. 10ECh. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - In Problems 712, find at least the first four...Ch. 8.4 - In Problems 1319, find at least the first four...Ch. 8.4 - In Problems 1319, find at least the first four...Ch. 8.4 - In Problems 1319, find at least the first four...Ch. 8.4 - Prob. 16ECh. 8.4 - In Problems 13-19, find at least the first four...Ch. 8.4 - In Problems 13-19, find at least the first four...Ch. 8.4 - In Problems 13-19, find at least the first four...Ch. 8.4 - To derive the general solution given by equations...Ch. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - Prob. 22ECh. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - Prob. 24ECh. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - In Problems 21-28, use the procedure illustrated...Ch. 8.4 - Prob. 28ECh. 8.4 - The equation (1x2)y2xy+n(n+1)y=0, where n is an...Ch. 8.4 - Aging Spring. As a spring ages, its spring...Ch. 8.4 - Aging Spring without Damping. In the mass-spring...Ch. 8.5 - Prob. 1ECh. 8.5 - Prob. 2ECh. 8.5 - Prob. 3ECh. 8.5 - Prob. 4ECh. 8.5 - Prob. 5ECh. 8.5 - Prob. 6ECh. 8.5 - Prob. 7ECh. 8.5 - Prob. 8ECh. 8.5 - Prob. 9ECh. 8.5 - Prob. 10ECh. 8.5 - Prob. 11ECh. 8.5 - Prob. 12ECh. 8.5 - Prob. 13ECh. 8.5 - Prob. 14ECh. 8.5 - Prob. 15ECh. 8.5 - Prob. 16ECh. 8.5 - In Problems 15-17, solve the given initial value...Ch. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.6 - In Problems 1-10, classify each singular point...Ch. 8.6 - Prob. 2ECh. 8.6 - Prob. 3ECh. 8.6 - Prob. 4ECh. 8.6 - Prob. 5ECh. 8.6 - Prob. 6ECh. 8.6 - Prob. 7ECh. 8.6 - Prob. 8ECh. 8.6 - Prob. 9ECh. 8.6 - Prob. 10ECh. 8.6 - Prob. 11ECh. 8.6 - In Problems 11-18, find the indicial equation and...Ch. 8.6 - In Problems 11-18, find the indicial equation and...Ch. 8.6 - In Problems 11-18, find the indicial equation and...Ch. 8.6 - In Problems 11-18, find the indicial equation and...Ch. 8.6 - In Problems 1118, find the indicial equation and...Ch. 8.6 - In Problems 1118, find the indicial equation and...Ch. 8.6 - In Problems 1118, find the indicial equation and...Ch. 8.6 - Prob. 19ECh. 8.6 - Prob. 20ECh. 8.6 - Prob. 21ECh. 8.6 - Prob. 22ECh. 8.6 - Prob. 23ECh. 8.6 - Prob. 24ECh. 8.6 - Prob. 25ECh. 8.6 - Prob. 26ECh. 8.6 - Prob. 27ECh. 8.6 - Prob. 28ECh. 8.6 - Prob. 29ECh. 8.6 - Prob. 30ECh. 8.6 - Prob. 31ECh. 8.6 - Prob. 32ECh. 8.6 - Prob. 33ECh. 8.6 - Prob. 34ECh. 8.6 - Prob. 35ECh. 8.6 - Prob. 36ECh. 8.6 - Prob. 37ECh. 8.6 - Prob. 38ECh. 8.6 - In Problems 39 and 40, try to use the method of...Ch. 8.6 - Prob. 40ECh. 8.6 - Prob. 41ECh. 8.6 - Prob. 42ECh. 8.6 - Prob. 43ECh. 8.6 - Prob. 44ECh. 8.6 - Prob. 45ECh. 8.6 - Prob. 46ECh. 8.6 - Prob. 47ECh. 8.7 - In Problems 1-14, find at least the first three...Ch. 8.7 - Prob. 2ECh. 8.7 - Prob. 3ECh. 8.7 - Prob. 4ECh. 8.7 - Prob. 5ECh. 8.7 - In Problems 1-14, find at least the first three...Ch. 8.7 - Prob. 7ECh. 8.7 - Prob. 8ECh. 8.7 - In Problems 1-14, find at least the first three...Ch. 8.7 - Prob. 10ECh. 8.7 - Prob. 11ECh. 8.7 - Prob. 12ECh. 8.7 - Prob. 13ECh. 8.7 - Prob. 14ECh. 8.7 - In Problems 15 and 16, determine whether the given...Ch. 8.7 - Prob. 16ECh. 8.7 - In Problems 17-20, find at least the first three...Ch. 8.7 - Prob. 18ECh. 8.7 - In Problems 17-20, find at least the first three...Ch. 8.7 - Prob. 20ECh. 8.7 - Prob. 21ECh. 8.7 - In Problem 21 consider a column with a rectangular...Ch. 8.7 - Prob. 23ECh. 8.7 - Prob. 24ECh. 8.7 - Prob. 25ECh. 8.7 - To obtain two linearly independent solutions to...Ch. 8.8 - In Problems 1-4, express a general solution to the...Ch. 8.8 - Prob. 2ECh. 8.8 - In Problems 1-4, express a general solution to the...Ch. 8.8 - In Problems 1-4, express a general solution to the...Ch. 8.8 - Prob. 5ECh. 8.8 - Prob. 6ECh. 8.8 - Prob. 7ECh. 8.8 - Prob. 8ECh. 8.8 - Prob. 9ECh. 8.8 - Prob. 10ECh. 8.8 - Show that the confluent hypergeometric equation...Ch. 8.8 - Prob. 12ECh. 8.8 - Prob. 13ECh. 8.8 - Prob. 14ECh. 8.8 - Prob. 15ECh. 8.8 - Prob. 16ECh. 8.8 - Prob. 17ECh. 8.8 - Prob. 18ECh. 8.8 - In Problems 19 and 20, a Bessel equation is given....Ch. 8.8 - Prob. 21ECh. 8.8 - Prob. 22ECh. 8.8 - Prob. 23ECh. 8.8 - Prob. 24ECh. 8.8 - Show that J1/2(x)=(2/x)1/2sinx and...Ch. 8.8 - The Bessel functions of order v=n+1/2, n any...Ch. 8.8 - Prob. 27ECh. 8.8 - Prob. 28ECh. 8.8 - Prob. 29ECh. 8.8 - Prob. 30ECh. 8.8 - Prob. 31ECh. 8.8 - To prove Rodriguess formula (52) for Legendre...Ch. 8.8 - Prob. 34ECh. 8.8 - Prob. 35ECh. 8.8 - Prob. 36ECh. 8.8 - The Hermite polynomials Hn(x) are polynomial...Ch. 8.8 - Prob. 38ECh. 8.8 - Prob. 39ECh. 8.8 - Reduction to Bessels Equation. The class of...Ch. 8.8 - a. Show that the substitution z(x)=xy(x) renders...Ch. 8.RP - Find the first four nonzero terms in the Taylor...Ch. 8.RP - Prob. 2RPCh. 8.RP - Find at least the first four nonzero terms in a...Ch. 8.RP - Prob. 4RPCh. 8.RP - Find at least the first four nonzero terms in a...Ch. 8.RP - Prob. 6RPCh. 8.RP - Use the method of Frobenius to find at least the...Ch. 8.RP - Find the indicial equation and its roots and state...Ch. 8.RP - Find at least the first three nonzero terms in the...
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