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Math Fundamentals of Differential Equations and Boundary Value Problems

In Problems 1-4 , find a general solution of the system x ′ ( t ) = A x ( t ) for the given matrix A . A = [ 2 − 4 2 − 2 ]

In Problems 1-4 , find a general solution of the system x ′ ( t ) = A x ( t ) for the given matrix A . A = [ 2 − 4 2 − 2 ]

Solution Summary: The author explains the general solution of the system xprime

Fundamentals of Differential Equations and Boundary Value Problems

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 9.6, Problem 1E

In Problems 1-4, find a general solution of the system x ′ ( t ) = A x ( t ) for the given matrix A .

A = [ 2 − 4 2 − 2 ]

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Chapter 9.5, Problem 50E

Chapter 9.6, Problem 2E

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1. Consider the linear equation system Ax = b with A = ( -9.8 19.8 -9.9 19.9 3) and b= = ( 1080 ) . In the following, you may use that the inverse of the matrix A is given by A-¹ = 1 = ( 19.9 -19.8 9.9 -9.8 8). (a) What is the solution x of Ax = b? (b) Let xx denote the solution of the perturbed equation (A+SA)(x+dx) = b for ||A|| 0.01. Show that the relative error is bounded by ||dx||∞ ||x||∞ ≤1.

determine the general solution to the system x′= Ax for the given matrix A

2. Suppose 3 2 1 A =3 4 2 3 2 2 (1) Find the LU factorization of the matrix A; (2) Using the LU factorization to solve the system of equations Ax =b where b=[4 5 s.

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. 2E Ch. 9.2 - Prob. 3E Ch. 9.2 - Prob. 4E Ch. 9.2 - Prob. 5E Ch. 9.2 - Prob. 6E Ch. 9.2 - Prob. 7E Ch. 9.2 - Prob. 8E Ch. 9.2 - Prob. 9E Ch. 9.2 - Prob. 10E Ch. 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. 13E Ch. 9.2 - Prob. 14E Ch. 9.3 - 9.3 Exercises Let A:=[2135] and B:=[1023] Find a...Ch. 9.3 - Prob. 2E Ch. 9.3 - Let A=[2411] and B=[21352] Find a AB. b A2=AA b...Ch. 9.3 - Prob. 4E Ch. 9.3 - Let A=[1223], B=[1011] and C=[1121] Find a AB. b...Ch. 9.3 - Prob. 6E Ch. 9.3 - 9.3 Exercises a. Show that if u and v are each n1...Ch. 9.3 - Prob. 8E Ch. 9.3 - Prob. 9E Ch. 9.3 - Prob. 10E 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 - 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. 18E Ch. 9.3 - Prob. 19E Ch. 9.3 - Prob. 20E Ch. 9.3 - Prob. 21E Ch. 9.3 - Prob. 22E Ch. 9.3 - Prob. 23E Ch. 9.3 - Prob. 24E Ch. 9.3 - Prob. 25E Ch. 9.3 - Prob. 26E Ch. 9.3 - Prob. 27E Ch. 9.3 - Prob. 28E Ch. 9.3 - Prob. 29E Ch. 9.3 - Prob. 30E Ch. 9.3 - Prob. 31E Ch. 9.3 - Prob. 32E Ch. 9.3 - Prob. 33E Ch. 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. 36E Ch. 9.3 - Prob. 37E Ch. 9.3 - Prob. 38E Ch. 9.3 - Prob. 39E Ch. 9.3 - Prob. 40E Ch. 9.3 - Prob. 41E Ch. 9.3 - Prob. 42E Ch. 9.3 - Prob. 43E Ch. 9.4 - In Problems 14, write the given system in the...Ch. 9.4 - Prob. 2E Ch. 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. 30E Ch. 9.4 - Prob. 31E Ch. 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. 34E Ch. 9.4 - Prob. 35E Ch. 9.4 - Prob. 36E Ch. 9.4 - Prob. 37E Ch. 9.4 - Prob. 38E Ch. 9.5 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - Prob. 2E 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 - In Problems 1-8, find the eigenvalues and...Ch. 9.5 - Prob. 8E Ch. 9.5 - Prob. 9E Ch. 9.5 - Prob. 10E 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 - 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. 36E Ch. 9.5 - Prob. 37E Ch. 9.5 - Prob. 38E Ch. 9.5 - a. Show that the matrix A=[211121221] has the...Ch. 9.5 - Prob. 40E Ch. 9.5 - Prob. 41E Ch. 9.5 - Prob. 42E Ch. 9.5 - Prob. 43E Ch. 9.5 - Prob. 44E Ch. 9.5 - Mixing Between Interconnected Tanks. Two tanks,...Ch. 9.5 - Prob. 46E Ch. 9.5 - Prob. 48E Ch. 9.5 - Stability.A homogeneous system x=Ax with constant...Ch. 9.5 - Prob. 50E 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 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. 21E Ch. 9.6 - Prob. 22E Ch. 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. 4E 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 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. 12E Ch. 9.7 - In Problems 11-16, use the variation of parameters...Ch. 9.7 - Prob. 14E Ch. 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. 24E Ch. 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. 28E Ch. 9.7 - Prob. 29E Ch. 9.7 - Prob. 30E Ch. 9.7 - Prob. 31E Ch. 9.7 - Prob. 32E Ch. 9.7 - Prob. 33E Ch. 9.7 - Prob. 34E Ch. 9.8 - In Problems 1-6, a show that the given matrix A...Ch. 9.8 - Prob. 2E Ch. 9.8 - Prob. 3E Ch. 9.8 - Prob. 4E Ch. 9.8 - Prob. 5E Ch. 9.8 - Prob. 6E Ch. 9.8 - In Problems 7-10, determine eAt by first finding a...Ch. 9.8 - Prob. 8E Ch. 9.8 - Prob. 9E Ch. 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. 18E Ch. 9.8 - Prob. 19E Ch. 9.8 - Prob. 20E Ch. 9.8 - Prob. 21E Ch. 9.8 - Prob. 22E Ch. 9.8 - Prob. 23E Ch. 9.8 - Prob. 24E Ch. 9.8 - Prob. 25E Ch. 9.8 - Prob. 26E 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 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. 6RP Ch. 9.RP - In Problems 7-10, find a general solution for the...Ch. 9.RP - Prob. 8RP Ch. 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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13. Find all value of h such that the augmented matrix (h+1) (h+2) 0 0 (h-2) (h-2) 0 0 (h² - 1) represents a system with: (a) no solutions; (b) a unique solution; (c) infinitely many solutions.
។ 4. Let A be an m x n matrix, with m

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