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 8.7, Problem 22E

In Problem 2 1 consider a column with a rectangular cross section with two sides constant and the other two changing linearly [see Figure 8 . 1 3 (b)]. In this case, n = 1 . Find at least the first four nonzero terms in the series expansion about x = 0 for a general solution to equation ( 4 5 ) when n = 1 .

21 .Buckling Columns. In the study of the buckling of a column whose cross section varies, one encounters the equation

x n y ( x ) + α 2 y ( x ) = 0 , x + 0 , ( 45 )

where x is related to the height above the ground and y is the deflection away from the vertical. The positive constant α depends on the rigidity of the column, its moment of inertia at the top, and the load. The positive integer n depends on the type of column. For example, when the column is a truncated cone [see Figure 8 . 1 3 (a) on page 4 7 4 ], we have n = 4 .

Chapter 8.7, Problem 22E, In Problem 21 consider a column with a rectangular cross section with two sides constant and the

a. Use the substitution x = t 1 to reduce ( 4 5 ) with n = 4 to the form

2 y t 2 + 2 t d y d t + α 2 y = 0 , t > 0 .

b. Find at least the first six nonzero terms in the series expansion about t = 0 for a general solution to the equation obtained in part (a).

c. Use the result of part (b) to give an expansion about x = for a general solution to ( 4 5 ) .

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Chapter 8 Solutions

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

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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