DIFF EQUAT W/BOUNDAR >PRINT UPGRADE<
9th Edition
ISBN: 9781337810906
Author: ZILL
Publisher: CENGAGE L
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Chapter 9.2, Problem 13E
(a)
To determine
The approximate value of velocity
(b)
To determine
To construct: The graph of the solution of the given initial value problem on the given interval
(c)
To determine
To solve: The initial value problem using the variable separable method and also find the actual value of
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Problem 11 (a) A tank is discharging water through an orifice at a depth of T
meter below the surface of the water whose area is A m². The
following are the values of a for the corresponding values of A:
A 1.257 1.390
x 1.50 1.65
1.520 1.650 1.809 1.962 2.123 2.295 2.462|2.650
1.80 1.95 2.10 2.25 2.40 2.55 2.70
2.85
Using the formula
-3.0
(0.018)T =
dx.
calculate T, the time in seconds for the level of the water to drop
from 3.0 m to 1.5 m above the orifice.
(b) The velocity of a train which starts from rest is given by the fol-
lowing table, the time being reckoned in minutes from the start
and the speed in km/hour:
| † (minutes) |2|4 6 8 10 12
14 16 18 20
v (km/hr) 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0
Estimate approximately the total distance ran in 20 minutes.
-
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?
Chapter 9 Solutions
DIFF EQUAT W/BOUNDAR >PRINT UPGRADE<
Ch. 9.1 - In Problems 110 use the improved Eulers method to...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - Prob. 5ECh. 9.1 - Prob. 6ECh. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - Prob. 9ECh. 9.1 - Prob. 10E
Ch. 9.1 - Prob. 11ECh. 9.1 - Prob. 13ECh. 9.1 - Prob. 14ECh. 9.1 - Prob. 15ECh. 9.1 - Prob. 16ECh. 9.1 - Consider the initial-value problem y = 2x 3y + 1,...Ch. 9.1 - Prob. 18ECh. 9.1 - Prob. 19ECh. 9.1 - Repeat Problem 19 using the improved Euler’s...Ch. 9.1 - Prob. 21ECh. 9.2 - Use the RK4 method with h = 0.1 to approximate...Ch. 9.2 - Prob. 2ECh. 9.2 - Prob. 3ECh. 9.2 - Prob. 4ECh. 9.2 - Prob. 5ECh. 9.2 - Prob. 6ECh. 9.2 - Prob. 7ECh. 9.2 - Prob. 8ECh. 9.2 - Prob. 9ECh. 9.2 - Prob. 10ECh. 9.2 - In Problems 312 use the RK4 method with h = 0.1 to...Ch. 9.2 - Prob. 12ECh. 9.2 - Prob. 13ECh. 9.2 - Consider the initial-value problem y′ = 2y, y(0) =...Ch. 9.2 - Prob. 17ECh. 9.2 - Consider the initial-value problem y′ = 2x – 3y +...Ch. 9.2 - Prob. 19ECh. 9.2 - Prob. 20ECh. 9.3 - Prob. 1ECh. 9.3 - Prob. 3ECh. 9.3 - Prob. 4ECh. 9.3 - Prob. 5ECh. 9.3 - Prob. 6ECh. 9.3 - Prob. 7ECh. 9.3 - Prob. 8ECh. 9.4 - Use Euler’s method to approximate y(0.2), where...Ch. 9.4 - Prob. 2ECh. 9.4 - Prob. 3ECh. 9.4 - Prob. 4ECh. 9.4 - Prob. 5ECh. 9.5 - Prob. 1ECh. 9.5 - Prob. 2ECh. 9.5 - Prob. 3ECh. 9.5 - Prob. 4ECh. 9.5 - Prob. 5ECh. 9.5 - In Problems 1-18 use Definition 7.1.1 to find ℒ{f...Ch. 9.5 - Prob. 7ECh. 9.5 - Prob. 8ECh. 9.5 - Prob. 9ECh. 9.5 - Prob. 10ECh. 9.5 - Prob. 11ECh. 9.5 - The electrostatic potential u between two...Ch. 9.5 - Consider the boundary-value problem y″ + xy = 0,...Ch. 9 - Prob. 1RECh. 9 - Prob. 2RECh. 9 - In Problems 1–4 construct a table comparing the...Ch. 9 - In Problems 1–4 construct a table comparing the...Ch. 9 - Prob. 5RECh. 9 - Prob. 6RECh. 9 - Prob. 7RECh. 9 - Prob. 8RE
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