Elementary Differential Equations
10th Edition
ISBN: 9780470458327
Author: William E. Boyce, Richard C. DiPrima
Publisher: Wiley, John & Sons, Incorporated
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Chapter 4.2, Problem 28P
To determine
The general solution of the higher order differential equation
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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, ...,
p-1
2
-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).
23
32
how come?
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
The set T is the subset of these residues exceeding
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
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?
Shading a Venn diagram with 3 sets: Unions, intersections, and...
The Venn diagram shows sets A, B, C, and the universal set U.
Shade (CUA)' n B on the Venn diagram.
U
Explanation
Check
A-
B
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Chapter 4 Solutions
Elementary Differential Equations
Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - Prob. 6PCh. 4.1 - In each of Problems 7 through 10, determine...Ch. 4.1 - Prob. 8PCh. 4.1 - In each of Problems 7 through 10, determine...Ch. 4.1 - Prob. 10P
Ch. 4.1 - In each of Problems 11 through 16, verify that the...Ch. 4.1 - In each of Problems 11 through 16, verify that the...Ch. 4.1 - In each of Problems 11 through 16, verify that the...Ch. 4.1 - In each of Problems 11 through 16, verify that the...Ch. 4.1 - In each of Problems 11 through 16, verify that the...Ch. 4.1 - In each of Problems 11 through 16, verify that the...Ch. 4.1 - Prob. 17PCh. 4.1 - Prob. 18PCh. 4.1 - Prob. 19PCh. 4.1 - Prob. 20PCh. 4.1 - Prob. 21PCh. 4.1 - Prob. 22PCh. 4.1 - Prob. 23PCh. 4.1 - Prob. 24PCh. 4.1 - Prob. 25PCh. 4.1 - Prob. 26PCh. 4.1 - Prob. 27PCh. 4.1 - Prob. 28PCh. 4.2 - In each of Problems 1 through 6, express the given...Ch. 4.2 - In each of Problems 1 through 6, express the given...Ch. 4.2 - In each of Problems 1 through 6, express the given...Ch. 4.2 - In each of Problems 1 through 6, express the given...Ch. 4.2 - In each of Problems 1 through 6, express the given...Ch. 4.2 - In each of Problems 1 through 6, express the given...Ch. 4.2 - Prob. 7PCh. 4.2 - Prob. 8PCh. 4.2 - Prob. 9PCh. 4.2 - Prob. 10PCh. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - Prob. 15PCh. 4.2 - Prob. 16PCh. 4.2 - Prob. 17PCh. 4.2 - Prob. 18PCh. 4.2 - Prob. 19PCh. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 11 through 28, find the...Ch. 4.2 - In each of Problems 29 through 36, find the...Ch. 4.2 - Prob. 31PCh. 4.2 - Prob. 32PCh. 4.2 - Prob. 33PCh. 4.2 - Prob. 34PCh. 4.2 - Prob. 35PCh. 4.2 - Prob. 36PCh. 4.2 - Prob. 37PCh. 4.2 - Prob. 38PCh. 4.2 - Prob. 39PCh. 4.2 - Prob. 40PCh. 4.3 - In each of Problems 1 through 8, determine the...Ch. 4.3 - In each of Problems 1 through 8, determine the...Ch. 4.3 - In each of Problems 1 through 8, determine the...Ch. 4.3 - In each of Problems 1 through 8, determine the...Ch. 4.3 - In each of Problems 1 through 8, determine the...Ch. 4.3 - In each of Problems 1 through 8, determine the...Ch. 4.3 - In each of Problems 1 through 8, determine the...Ch. 4.3 - In each of Problems 1 through 8, determine the...Ch. 4.3 - In each of Problems 9 through 12, find the...Ch. 4.3 - In each of Problems 9 through 12, find the...Ch. 4.3 - In each of Problems 9 through 12, find the...Ch. 4.3 - In each of Problems 9 through 12, find the...Ch. 4.3 - In each of Problems 13 through 18, determine a...Ch. 4.3 - In each of Problems 13 through 18, determine a...Ch. 4.3 - In each of Problems 13 through 18, determine a...Ch. 4.3 - In each of Problems 13 through 18, determine a...Ch. 4.3 - In each of Problems 13 through 18, determine a...Ch. 4.3 - In each of Problems 13 through 18, determine a...Ch. 4.3 - Prob. 19PCh. 4.3 - Show that linear differential operators with...Ch. 4.4 - Prob. 1PCh. 4.4 - In each of Problems 1 through 6, use the method of...Ch. 4.4 - In each of Problems 1 through 6, use the method of...Ch. 4.4 - In each of Problems 1 through 6, use the method of...Ch. 4.4 - In each of Problems 1 through 6, use the method of...Ch. 4.4 - In each of Problems 1 through 6, use the method of...Ch. 4.4 - In each of Problems 7 and 8, find the general...Ch. 4.4 - In each of Problems 7 and 8, find the general...Ch. 4.4 - In each of Problems 9 through 12, find the...Ch. 4.4 - In each of Problems 9 through 12, find the...Ch. 4.4 - In each of Problems 9 through 12, find the...Ch. 4.4 - In each of Problems 9 through 12, find the...Ch. 4.4 - Given that x, x2, and 1/x are solutions of the...Ch. 4.4 - Find a formula involving integrals for a...Ch. 4.4 - Find a formula involving integrals for a...Ch. 4.4 - Find a formula involving integrals for a...Ch. 4.4 - Find a formula involving integrals for a...
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