Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
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Chapter 9.2, Problem 1E
In Problems 1-11, find all solutions to the system using the Gauss-Jordan elimination algorithm.
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The set of all 3 x 3 upper triangular matrices
6) Determine whether each of the following sets, together with the standard
operations, is a vector space. If it is, then simply write 'Vector space'. You do not
have to prove all ten vector space axioms. If it is not, then identify one of the ten
vector space axioms with its number in the attached sheet that fails and also show
that how it fails.
a) The set of all polynomials of degree four or less.
b) The set of all 2 x 2 singular matrices.
c) The set {(x, y) : x ≥ 0, y is a real number}.
d) C[0,1], the set of all continuous functions defined on the interval [0,1].
7) Given u = (-2,1,1) and v = (4,2,0) are two vectors in R³-space. Find u xv and
show that it is orthogonal to both u and v.
8) a) Find the equation of the least squares regression line for the data points
below.
(-2,0), (0,2), (2,2)
b) Graph the points and the line that you found from a) on the same Cartesian
coordinate plane.
1. A consumer group claims that the mean annual consumption of cheddar cheese by a person in
the United States is at most 10.3 pounds. A random sample of 100 people in the United States has
a mean annual cheddar cheese consumption of 9.9 pounds. Assume the population standard
deviation is 2.1 pounds. At a = 0.05, can you reject the claim? (Adapted from U.S. Department of
Agriculture)
State the hypotheses:
Calculate the test statistic:
Calculate the P-value:
Conclusion (reject or fail to reject Ho):
2. The CEO of a manufacturing facility claims that the mean workday of the company's assembly
line employees is less than 8.5 hours. A random sample of 25 of the company's assembly line
employees has a mean workday of 8.2 hours. Assume the population standard deviation is 0.5
hour and the population is normally distributed. At a = 0.01, test the CEO's claim.
State the hypotheses:
Calculate the test statistic:
Calculate the P-value:
Conclusion (reject or fail to reject Ho):
Statistics
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1) a) Find a matrix P such that PT AP orthogonally diagonalizes the following matrix
A.
= [{² 1]
A =
b) Verify that PT AP gives the correct diagonal form.
2
01
-2
3
2) Given the following matrices A =
-1
0
1] an
and B =
0
1
-3
2
find the following matrices:
a) (AB) b) (BA)T
3) Find the inverse of the following matrix A using Gauss-Jordan elimination or
adjoint of the matrix and check the correctness of your answer (Hint: AA¯¹ = I).
[1 1 1
A = 3 5 4
L3 6 5
4) Solve the following system of linear equations using any one of Cramer's Rule,
Gaussian Elimination, Gauss-Jordan Elimination or Inverse Matrix methods and
check the correctness of your answer.
4x-y-z=1
2x + 2y + 3z = 10
5x-2y-2z = -1
5) a) Describe the zero vector and the additive inverse of a vector in the vector
space, M3,3.
b) Determine if the following set S is a subspace of M3,3 with the standard
operations. Show all appropriate supporting work.
Chapter 9 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 1 -6, express the given system of...Ch. 9.1 - In Problems 7 -10, express the given higher-order...Ch. 9.1 - In Problems 7 -10, express the given higher-order...Ch. 9.1 - In Problems 7 -10, express the given higher-order...Ch. 9.1 - In Problems 7 -10, express the given higher-order...
Ch. 9.1 - In Problems 11 -13, express the given system of...Ch. 9.1 - In Problems 11 -13, express the given system of...Ch. 9.1 - In Problems 11 -13, express the given system of...Ch. 9.2 - In Problems 1-11, find all solutions to the system...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 1-11, find all solutions to the system...Ch. 9.2 - Use the Gauss-Jordan elimination algorithm to show...Ch. 9.2 - Prob. 13ECh. 9.2 - Prob. 14ECh. 9.3 - 9.3 Exercises Let A:=[2135] and B:=[1023] Find a...Ch. 9.3 - Prob. 2ECh. 9.3 - Let A=[2411] and B=[21352] Find a AB. b A2=AA b...Ch. 9.3 - Prob. 4ECh. 9.3 - Let A=[1223], B=[1011] and C=[1121] Find a AB. b...Ch. 9.3 - Prob. 6ECh. 9.3 - 9.3 Exercises a. Show that if u and v are each n1...Ch. 9.3 - Prob. 8ECh. 9.3 - Prob. 9ECh. 9.3 - Prob. 10ECh. 9.3 - In Problem 9-14, use the method of Example 1 to...Ch. 9.3 - In Problem 9-14, use the method of Example 1 to...Ch. 9.3 - In Problem 9-14, use the method of Example 1 to...Ch. 9.3 - In Problem 9-14, use the method of Example 1 to...Ch. 9.3 - Prove that if xp satisfy Axp=b, then every...Ch. 9.3 - Let A=[211121112] a. Show that A is singular. b....Ch. 9.3 - In Problems 17-20, find the matrix X1(t) whose...Ch. 9.3 - Prob. 18ECh. 9.3 - Prob. 19ECh. 9.3 - Prob. 20ECh. 9.3 - Prob. 21ECh. 9.3 - Prob. 22ECh. 9.3 - Prob. 23ECh. 9.3 - Prob. 24ECh. 9.3 - Prob. 25ECh. 9.3 - Prob. 26ECh. 9.3 - Prob. 27ECh. 9.3 - Prob. 28ECh. 9.3 - Prob. 29ECh. 9.3 - Prob. 30ECh. 9.3 - Prob. 31ECh. 9.3 - Prob. 32ECh. 9.3 - Prob. 33ECh. 9.3 - 9.3 Exercises Find dx/dt for the given matrix...Ch. 9.3 - 9.3 Exercises Verify that the given vector...Ch. 9.3 - Prob. 36ECh. 9.3 - Prob. 37ECh. 9.3 - Prob. 38ECh. 9.3 - Prob. 39ECh. 9.3 - Prob. 40ECh. 9.3 - Prob. 41ECh. 9.3 - Prob. 42ECh. 9.3 - Prob. 43ECh. 9.4 - In Problems 14, write the given system in the...Ch. 9.4 - Prob. 2ECh. 9.4 - In Problems 14, write the given system in the...Ch. 9.4 - In Problems 14, write the given system in the...Ch. 9.4 - In Problems 5-8, rewrite the given scalar equation...Ch. 9.4 - In Problems 5-8, rewrite the given scalar equation...Ch. 9.4 - In Problems 5-8, rewrite the given scalar equation...Ch. 9.4 - In Problems 5-8, rewrite the given scalar equation...Ch. 9.4 - In Problems 9-12, write the given system as a set...Ch. 9.4 - In Problems 9-12, write the given system as a set...Ch. 9.4 - In Problems 9-12, write the given system as a set...Ch. 9.4 - In Problems 9-12, write the given system as a set...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problems 13-19, determine whether the given...Ch. 9.4 - In Problem 13-19, determine whether the given...Ch. 9.4 - Let...Ch. 9.4 - In Problems 21-24, the given vector functions are...Ch. 9.4 - In Problems 21-24, the given vector functions are...Ch. 9.4 - In Problems 21-24, the given vector functions are...Ch. 9.4 - In Problems 21-24, the given vector functions are...Ch. 9.4 - Verify that the vector functions x1=[etet] and...Ch. 9.4 - Verify that the vector functions...Ch. 9.4 - Prove that the operator define by L[x]:=xAx, where...Ch. 9.4 - Let X(t) be a fundamental matrix for the system...Ch. 9.4 - In Problem 29-30, verify that X(t) is a...Ch. 9.4 - Prob. 30ECh. 9.4 - Prob. 31ECh. 9.4 - Abels Formula. If x1,,xn are any n solutions to...Ch. 9.4 - Using Abels formula Problem 32, confirm that the...Ch. 9.4 - Prob. 34ECh. 9.4 - Prob. 35ECh. 9.4 - Prob. 36ECh. 9.4 - Prob. 37ECh. 9.4 - Prob. 38ECh. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - Prob. 2ECh. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - Prob. 8ECh. 9.5 - Prob. 9ECh. 9.5 - Prob. 10ECh. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - In Problems 11-16, find a general solution of the...Ch. 9.5 - Consider the system x (t)=Ax(t),t0 with A=[1331]...Ch. 9.5 - Consider the system x(t)=Ax(t),t0 with A=[2112] a....Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 19-24, find a fundamental matrix for...Ch. 9.5 - In Problems 31-34, solve the given initial value...Ch. 9.5 - In Problems 31-34, solve the given initial value...Ch. 9.5 - In Problems 31-34, solve the given initial value...Ch. 9.5 - In Problems 31-34, solve the given initial value...Ch. 9.5 - a. Show that the matrix A=[1143] has the repeated...Ch. 9.5 - Prob. 36ECh. 9.5 - Prob. 37ECh. 9.5 - Prob. 38ECh. 9.5 - a. Show that the matrix A=[211121221] has the...Ch. 9.5 - Prob. 40ECh. 9.5 - Prob. 41ECh. 9.5 - Prob. 42ECh. 9.5 - Prob. 43ECh. 9.5 - Prob. 44ECh. 9.5 - Mixing Between Interconnected Tanks. Two tanks,...Ch. 9.5 - Prob. 46ECh. 9.5 - Prob. 48ECh. 9.5 - Stability.A homogeneous system x=Ax with constant...Ch. 9.5 - Prob. 50ECh. 9.6 - In Problems 1-4, find a general solution of the...Ch. 9.6 - In Problems 1-4, find a general solution of the...Ch. 9.6 - In Problems 1-4, find a general solution of the...Ch. 9.6 - In Problems 1-4, find a general solution of the...Ch. 9.6 - In Problems 5-9, find a fundamental matrix for the...Ch. 9.6 - In Problems 5-9, find a fundamental matrix for the...Ch. 9.6 - In Problems 5-9, find a fundamental matrix for the...Ch. 9.6 - In Problems 5-9, find a fundamental matrix for the...Ch. 9.6 - 9.6 Exercises Find a fundamental matrix for the...Ch. 9.6 - In Problems 13 and 14, find the solution to the...Ch. 9.6 - In Problems 13 and 14, find the solution to the...Ch. 9.6 - Show that x1(t) and x2(t) given by equations 4 and...Ch. 9.6 - Show that x1(t) and x2(t) given by equations 4 and...Ch. 9.6 - In Problems 17 and 18, use the results of Problem...Ch. 9.6 - In Problems 17 and 18, use the results of Problem...Ch. 9.6 - For the coupled mass-spring system governed by...Ch. 9.6 - For the coupled mas-spring system governed by...Ch. 9.6 - Prob. 21ECh. 9.6 - Prob. 22ECh. 9.6 - Stability: In Problem 49 of Exercises 9.5, page...Ch. 9.6 - a. a For Example 1, page 535, verify that...Ch. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - Prob. 4ECh. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - In Problems 1-6, use the method of undetermined...Ch. 9.7 - In Problems 7-10, use the method of undetermined...Ch. 9.7 - In Problems 7-10, use the method of undetermined...Ch. 9.7 - In Problems 7-10, use the method of undetermined...Ch. 9.7 - In Problems 7-10, use the method of undetermined...Ch. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - Prob. 12ECh. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - Prob. 14ECh. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - Find the solution to the given system that...Ch. 9.7 - 9.7 Exercises In Problems 21 and 22, find the...Ch. 9.7 - Using matrix algebra techniques and method of...Ch. 9.7 - Prob. 24ECh. 9.7 - To find a general solution to the system...Ch. 9.7 - For the system of Problem 25, we found that a...Ch. 9.7 - .Find a general solution of the system...Ch. 9.7 - Prob. 28ECh. 9.7 - Prob. 29ECh. 9.7 - Prob. 30ECh. 9.7 - Prob. 31ECh. 9.7 - Prob. 32ECh. 9.7 - Prob. 33ECh. 9.7 - Prob. 34ECh. 9.8 - In Problems 1-6, a show that the given matrix A...Ch. 9.8 - Prob. 2ECh. 9.8 - Prob. 3ECh. 9.8 - Prob. 4ECh. 9.8 - Prob. 5ECh. 9.8 - Prob. 6ECh. 9.8 - In Problems 7-10, determine eAt by first finding a...Ch. 9.8 - Prob. 8ECh. 9.8 - Prob. 9ECh. 9.8 - In Problems 7-10, determine eAt by first finding a...Ch. 9.8 - In Problems 11 and 12, determine eAt by using...Ch. 9.8 - In Problems 11 and 12, determine eAt by using...Ch. 9.8 - In Problems 17-20, use the generalized...Ch. 9.8 - Prob. 18ECh. 9.8 - Prob. 19ECh. 9.8 - Prob. 20ECh. 9.8 - Prob. 21ECh. 9.8 - Prob. 22ECh. 9.8 - Prob. 23ECh. 9.8 - Prob. 24ECh. 9.8 - Prob. 25ECh. 9.8 - Prob. 26ECh. 9.RP - In Problems 1-4, find a general solution for the...Ch. 9.RP - In Problems 1-4, find a general solution for the...Ch. 9.RP - In Problems 1-4, find a general solution for the...Ch. 9.RP - In Problems 1-4, find a general solution for the...Ch. 9.RP - In Problems 5 and 6, find a fundamental matrix for...Ch. 9.RP - Prob. 6RPCh. 9.RP - In Problems 7-10, find a general solution for the...Ch. 9.RP - Prob. 8RPCh. 9.RP - In Problems 7-10, find a general solution for the...Ch. 9.RP - In Problems 7-10, find a general solution for the...Ch. 9.RP - In Problems 11 and 12, solve the given initial...Ch. 9.RP - In Problems 11 and 12, solve the given initial...Ch. 9.RP - In Problems 13 and 14, find a general solution for...Ch. 9.RP - In Problems 13 and 14, find a general solution for...Ch. 9.RP - In Problems 15 and 16, find the fundamental matrix...Ch. 9.RP - In Problems 15 and 16, find the fundamental matrix...
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