4th Edition

ISBN: 9780495561989

Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall

Publisher: Cengage Learning

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Chapter 3.6, Problem 17E

In Exercises 13-20, consider harmonic oscillators with mass m, spring constant k, and damping coefficient b. For the values specified,
(a) write the second-order differential equation and the corresponding first-order system;
(b) find the eigenvalues and eigenvectors of the linear system;
(c) classify the oscillator (as underdamped, overdamped, critically damped, or undamped) and, when appropriate, give the natural period;
(d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and
(e) sketch the y(t) - and v(t) -graphs of the solution with the given initial condition.

17. m = 2 , k = 1 , b = 3 , with initial conditions y ( 0 ) = 0 , v ( 0 ) = 3 A

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Chapter 3.6, Problem 16E

Chapter 3.6, Problem 18E

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Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., 2 p-1 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. 23 The set T is the subset of these residues exceeding 2° So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1. how come?

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