Differential Equations with Boundary-Value Problems (MindTap Course List)
9th Edition
ISBN: 9781305965799
Author: Dennis G. Zill
Publisher: Cengage Learning
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Chapter 12.1, Problem 1E
In Problems 1–16 use separation of variables to find, if possible, product solutions for the given partial differential equation.
1.
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5. [-/1 Points]
DETAILS
MY NOTES
SESSCALCET2 6.5.AE.003.
y
y= ex²
0
Video Example
x
EXAMPLE 3
(a) Use the Midpoint Rule with n = 10 to approximate the integral
कर
L'ex²
dx.
(b) Give an upper bound for the error involved in this approximation.
SOLUTION
8+2
1
L'ex² d
(a) Since a = 0, b = 1, and n = 10, the Midpoint Rule gives the following. (Round your answer to six decimal places.)
dx Ax[f(0.05) + f(0.15) + ... + f(0.85) + f(0.95)]
0.1 [0.0025 +0.0225
+
+ e0.0625 + 0.1225
e0.3025 + e0.4225
+ e0.2025
+
+ e0.5625 €0.7225 +0.9025]
The figure illustrates this approximation.
(b) Since f(x) = ex², we have f'(x)
=
0 ≤ f'(x) =
< 6e.
ASK YOUR TEACHER
and f'(x) =
Also, since 0 ≤ x ≤ 1 we have x² ≤
and so
Taking K = 6e, a = 0, b = 1, and n = 10 in the error estimate, we see that an upper bound for the error is as follows. (Round your final
answer to five decimal places.)
6e(1)3
e
24(
=
≈
1. Consider the following preference ballots:
Number of voters
Rankings 6 5 4 2
1st choice A DCB
DC
2nd choice B B D
3rd choice DCBD
4th choice CA
AAA
For each of the four voting systems we have studied, determine who would win the election
in each case. (Remember: For plurality with runoff, all but the top two vote-getters are
simultaneously eliminated at the end of round 1.)
Practice
k Help
ises
A
96
Anewer The probability that you get a sum of at least 10 is
Determine the number of ways that the specified event can occur when
two number cubes are rolled.
1. Getting a sum of 9 or 10
3. Getting a sum less than 5
2. Getting a sum of 6 or 7
4. Getting a sum that is odd
Tell whether you would use the addition principle or the multiplication
principle to determine the total number of possible outcomes for the
situation described.
5. Rolling three number cubes
6. Getting a sum of 10 or 12 after rolling three number cubes
A set of playing cards contains four groups of cards designated by color
(black, red, yellow, and green) with cards numbered from 1 to 14 in each
group. Determine the number of ways that the specified event can occur
when a card is drawn from the set.
7. Drawing a 13 or 14
9. Drawing a number less than 4
8. Drawing a yellow or green card
10. Drawing a black, red, or green car
The spinner is divided into equal parts.
Find the specified…
Chapter 12 Solutions
Differential Equations with Boundary-Value Problems (MindTap Course List)
Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...
Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 18ECh. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 20ECh. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 22ECh. 12.1 - Prob. 23ECh. 12.1 - Prob. 24ECh. 12.1 - Prob. 25ECh. 12.1 - Prob. 26ECh. 12.1 - In Problems 27 and 28 show that the given partial...Ch. 12.1 - In Problems 27 and 28 show that the given partial...Ch. 12.1 - Verify that each of the products u = XY in (3),...Ch. 12.1 - Prob. 30ECh. 12.1 - Prob. 31ECh. 12.1 - Prob. 32ECh. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - Prob. 10ECh. 12.2 - In Problems 11 and 12 set up the boundary-value...Ch. 12.2 - In Problems 11 and 12 set up the boundary-value...Ch. 12.3 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.3 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.3 - Find the temperature u(x, t) in a rod of length L...Ch. 12.3 - Solve Problem 3 if L = 2 and f(x)={x,0x10,1x2.Ch. 12.3 - Suppose heat is lost from the lateral surface of a...Ch. 12.3 - Solve Problem 5 if the ends x = 0 and x = L are...Ch. 12.3 - A thin wire coinciding with the x-axis on the...Ch. 12.3 - Find the temperature u(x, t) for the...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - Prob. 11ECh. 12.4 - A model for the motion of a vibrating string whose...Ch. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - The transverse displacement u(x, t) of a vibrating...Ch. 12.4 - Prob. 19ECh. 12.4 - The vertical displacement u(x, t) of an infinitely...Ch. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 1–10 solve Laplace’s equation (1) for...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 1–10 solve Laplace’s equation (1) for...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - Prob. 10ECh. 12.5 - In Problems 11 and 12 solve Laplaces equation (1)...Ch. 12.5 - In Problems 11 and 12 solve Laplaces equation (1)...Ch. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - In Problems 15 and 16 use the superposition...Ch. 12.5 - In Problems 15 and 16 use the superposition...Ch. 12.5 - Prob. 18ECh. 12.5 - Solve the Neumann problem for a rectangle:...Ch. 12.5 - Prob. 20ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 3ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 6ECh. 12.6 - Prob. 7ECh. 12.6 - Prob. 8ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - In Problems 13-16 proceed as in Example 2 to solve...Ch. 12.6 - Prob. 15ECh. 12.6 - In Problems 13-16 proceed as in Example 2 to solve...Ch. 12.6 - Prob. 17ECh. 12.6 - Prob. 18ECh. 12.6 - Prob. 19ECh. 12.6 - Prob. 20ECh. 12.7 - In Example 1 find the temperature u(x, t) when the...Ch. 12.7 - Prob. 2ECh. 12.7 - Find the steady-state temperature for a...Ch. 12.7 - Prob. 4ECh. 12.7 - Prob. 5ECh. 12.7 - Prob. 6ECh. 12.7 - Prob. 7ECh. 12.7 - Prob. 8ECh. 12.7 - Prob. 9ECh. 12.7 - Prob. 10ECh. 12.8 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.8 - Prob. 2ECh. 12.8 - Prob. 3ECh. 12.8 - In Problems 3 and 4 solve the wave equation (2)...Ch. 12.8 - Prob. 5ECh. 12.8 - Prob. 6ECh. 12 - Use separation of variables to find product...Ch. 12 - Use separation of variables to find product...Ch. 12 - Find a steady-state solution (x) of the...Ch. 12 - Give a physical interpretation for the boundary...Ch. 12 - At t = 0 a string of unit length is stretched on...Ch. 12 - Prob. 6RECh. 12 - Find the steady-state temperature u(x, y) in the...Ch. 12 - Find the steady-state temperature u(x, y) in the...Ch. 12 - Prob. 9RECh. 12 - Find the temperature u(x, t) in the infinite plate...Ch. 12 - Prob. 11RECh. 12 - Solve the boundary-value problem 2ux2+sinx=ut, 0 ...Ch. 12 - Prob. 13RECh. 12 - The concentration c(x, t) of a substance that both...Ch. 12 - Prob. 15RECh. 12 - Solve Laplaces equation for a rectangular plate...Ch. 12 - Prob. 17RECh. 12 - Prob. 18RECh. 12 - Prob. 19RECh. 12 - If the four edges of the rectangular plate in...
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