Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
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Chapter 6.6, Problem 8P
In each of Problems 2 through 9, find the general solution of the given system. Calculations are greatly facilitated by using a computer algebra system.
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In each of Problems 1 through 8, express the general solution of the given system of equations in terms of real-valued functions:
1-i
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Chapter 6 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Ch. 6.1 - If and Find :
Ch. 6.1 - Verify that x=et(684)+2e2t(011) satisfies...Ch. 6.1 - Verify that =(ete2te3t4ete2t2e3tete2te3t)...Ch. 6.1 - In each of Problems through, transform equation...Ch. 6.1 - In each of Problems 4 through 9, transform...Ch. 6.1 - In each of Problems through, transform equation...Ch. 6.1 - In each of Problems through, transform equation...Ch. 6.1 - In each of Problems 4 through 9, transform...Ch. 6.1 - In each of Problems 4 through 9, transform...Ch. 6.1 - Derive the differential equationsfor x1(t) and...
Ch. 6.1 - Determine the matrix K and input g(t) if the (23)...Ch. 6.1 - Find a system of first order linear differential...Ch. 6.1 - An initial amount of tracer (such as a dye or a...Ch. 6.1 - Using matrix notation, show that the system of...Ch. 6.1 - Consider the plant equation (26) for the control...Ch. 6.2 - In each of problems 1 through 6, determine...Ch. 6.2 - In each of problems 1 through 6, determine...Ch. 6.2 - In each of problems 1 through 6, determine...Ch. 6.2 - In each of problems 1 through 6, determine...Ch. 6.2 - In each of problems through ,determine intervals...Ch. 6.2 - In each of problems 1 through 6, determine...Ch. 6.2 - Consider the vectors x1(t)=(et2etet),...Ch. 6.2 - Determine whether
, ,
form a fundamental set...Ch. 6.2 - Determine whether x1(t)=et(101), x2(t)=et(141),...Ch. 6.2 - In section it was shown that if and are...Ch. 6.2 - In each of problems 11 through 16, verify that the...Ch. 6.2 - In each of problems 11 through 16, verify that the...Ch. 6.2 - In each of problems 11 through 16, verify that the...Ch. 6.2 - In each of problems through , verify that the...Ch. 6.2 - In each of problems through , verify that the...Ch. 6.2 - In each of problems through , verify that the...Ch. 6.2 -
Verify that the differential operator defined by...Ch. 6.3 - In each of problems 1 through 8, find the general...Ch. 6.3 - In each of problems through ,find the general...Ch. 6.3 - In each of problems through ,find the general...Ch. 6.3 - In each of problems through ,find the general...Ch. 6.3 - In each of problems 1 through 8, find the general...Ch. 6.3 - In each of problems 1 through 8, find the general...Ch. 6.3 - In each of problems through ,find the general...Ch. 6.3 - In each of problems 1 through 8, find the general...Ch. 6.3 - In each of problems through , solve the given...Ch. 6.3 - In each of problems 9 through 12, solve the given...Ch. 6.3 - In each of problems 9 through 12, solve the given...Ch. 6.3 - In each of problems 9 through 12, solve the given...Ch. 6.3 - Using the rate equations (20) through (22),...Ch. 6.3 - Diffusion on a One-dimensional Lattice with an...Ch. 6.3 - Find constant vectors and such that the...Ch. 6.3 - Find constant vectors and such that the...Ch. 6.3 - A radioactive substance having decay rate ...Ch. 6.3 - For each of the matrices in Problems 18 through...Ch. 6.3 - For each of the matrices in Problems through ,...Ch. 6.3 - For each of the matrices in Problems through ,...Ch. 6.3 - For each of the matrices in Problems through ,...Ch. 6.3 - For each of the matrices in Problems 18 through...Ch. 6.3 - For each of the matrices in Problems through ,...Ch. 6.4 - In each of problems through , express the...Ch. 6.4 - In each of problems 1 through 8, express the...Ch. 6.4 - In each of problems through , express the...Ch. 6.4 - In each of problems through , express the...Ch. 6.4 - In each of problems 1 through 8, express the...Ch. 6.4 - In each of problems 1 through 8, express the...Ch. 6.4 - In each of problems through , express the...Ch. 6.4 - In each of problems through , express the...Ch. 6.4 -
(a) Find constant vectors and such that the...Ch. 6.4 -
(a) Find constant vectors and such that the...Ch. 6.4 - In this problem, we indicate how to show that...Ch. 6.4 - Consider the two-mass, three-spring system of...Ch. 6.4 - Consider the two-mass, three-spring system whose...Ch. 6.4 - Consider the two-mass, three-spring system whose...Ch. 6.4 - For each of the matrices in problem 15 through 18...Ch. 6.4 -
For each of the matrices in problem through use...Ch. 6.4 - For each of the matrices in problem 15 through 18...Ch. 6.4 - For each of the matrices in problem 15 through 18...Ch. 6.5 - In each of problem through , find a fundamental...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem through , find a fundamental...Ch. 6.5 - In each of problem through , find a fundamental...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem through , find a fundamental...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - In each of problem 1 through 14, find a...Ch. 6.5 - Solve the initial value problem...Ch. 6.5 - Solve the initial value problem...Ch. 6.5 - In each of Problems 17 through 20, use the method...Ch. 6.5 - In each of Problems through , use the method of...Ch. 6.5 - In each of Problems 17 through 20, use the method...Ch. 6.5 - In each of Problems 17 through 20, use the method...Ch. 6.5 - Consider an oscillator satisfying the initial...Ch. 6.5 - The matrix of coefficients for the system of...Ch. 6.5 - Assume that the real nn matrix A has n linearly...Ch. 6.5 - The Method of Successive Approximations. Consdier...Ch. 6.6 - Assuming that is a fundamental matrix for , show...Ch. 6.6 - In each of Problems 2 through 9, find the general...Ch. 6.6 - In each of Problems 2 through 9, find the general...Ch. 6.6 - In each of Problems 2 through 9, find the general...Ch. 6.6 - In each of Problems 2 through 9, find the general...Ch. 6.6 - In each of Problems 2 through 9, find the general...Ch. 6.6 - In each of Problems 2 through 9, find the general...Ch. 6.6 - In each of Problems 2 through 9, find the general...Ch. 6.6 - In each of Problems 2 through 9, find the general...Ch. 6.6 - Diffusion of particles on a lattice with...Ch. 6.6 - Find numerical approximations to the initial value...Ch. 6.6 - The equations presented in Section 6.1 for...Ch. 6.6 - When viscous damping forces are included and the...Ch. 6.6 - Undetermined Coefficients. For each of the...Ch. 6.6 - Undetermined Coefficients. For each of the...Ch. 6.6 - Undetermined Coefficients. For each of the...Ch. 6.7 - In each of Problems 1 through 8, find a...Ch. 6.7 - In each of Problems 1 through 8, find a...Ch. 6.7 - In each of Problems 1 through 8, find a...Ch. 6.7 - In each of Problems 1 through 8, find a...Ch. 6.7 - In each of Problems 1 through 8, find a...Ch. 6.7 - In each of Problems 1 through 8, find a...Ch. 6.7 - In each of Problems 1 through 8, find a...Ch. 6.7 - In each of Problems 1 through 8, find a...Ch. 6.7 - In each of Problems 9 and 10, find the solution of...Ch. 6.7 - In each of Problems 9 and 10, find the solution of...Ch. 6.7 - In each of Problems 11and12, find the solution of...Ch. 6.7 - In each of Problems 11 and 12, find the solution...Ch. 6.P1 - The Undamped Building. (a) Show that...Ch. 6.P1 - The Building with Damping Devices. In addition to...Ch. 6.P1 - A majority of the buildings that collapsed during...Ch. 6.P2 - Derive the system of equations (1) by applying...Ch. 6.P2 - Find the eigenvalues and eigenvectors of the...Ch. 6.P2 - From the normal mode representation of the...Ch. 6.P2 - Repeat Problem 2 for a system of four masses...Ch. 6.P2 - Find the rank of the controllability matrix for...Ch. 6.P2 - Find the rank of the controllability matrix for...Ch. 6.P2 - Prove the Cayley–Hamilton theorem for the special...Ch. 6.P2 - A symmetric matrix is said to be negative definite...Ch. 6.P2 - For the three-mass system, find a scalar control...
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- In each of Problems (2) through (9), find the general solution of the given system. Calculations are greatly facilitated by using a computer algebra system. 2.x² - (²-2) x + ( ) X' = 3arrow_forwardIf a and ß are the roots of the equation x² – 3x + 4 = 0, the equation whose roots are a? + a + 1 and B² + B + 1 is (a) x' - 6x + 19 = 0 (b) х? + 6х - 37 %3D 0 (c) x' + 6x - 19 = 0 (d) x² - 6x + 37 = 0 If a, and B, are the roots of the equation 3x? - 2x – 5 = 0 and a, and B, are the roots of the equation 2x² + x - 7 = 0, the equation whose roots are (a,a, + B,B,) and (a,ß, + a,B,) is (a) 36x? – 12x + 911 = 0 (b) 9х* - 91х + 11 %3D0 (c) 36x? + 12x - 911 = 0 (d) 9x² + 91x – 19 = 0 3x-2 If a and B are the roots of the equation 7X- 3 5x 1 = 4, 1 isarrow_forwardSolve part b of the problemarrow_forward
- Consider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds x₁(t) and x₂(t) of salt in tanks A and B, respectively, at time t. dx₁1 dt dx1 dt pure water 3 gal/min mixture 4 gal/min This system is described by the system of equations 1 50 2 25 dx2 dt 2 25 2 25*1 1 + -X2 mixture 1 gal/min -X2 B with initial conditions x₁(0) = 25, x₂(0) = 0 (see (3) and the surrounding discussion on mixtures on page 107). What is the system of differential equations if, instead of pure water, a brine solution containing 3 pounds of salt per gallon is pumped into tank A? -2 25*1 + 50x2+6× dx2 2 - (25)*₁- (25)×₂2 - = x2 dt…arrow_forwardConsider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds x₁(t) and x₂(t) of salt in tanks A and B, respectively, at time t. dx₁ dt dx₂ dt mixture 4 gal/min This system is described by the system of equations 1 50 2 2 25 2 dx₁ dt dx2 dt = = pure water 3 gal/min = 2 25 2 110 25 A + 1 1 mixture 1 gal/min B with initial conditions x₁(0) = 25, x₂(0) = 0 (see (3) and the surrounding discussion on mixtures on page 107). What is the system of differential equations if, instead of pure water, a brine solution containing 4 pounds of salt per gallon is pumped into tank A? mixture 3 gal/minarrow_forwardQuestions 23 and 24 pertain to the following situation. From 2004 to 2013, the average salary, y, in thousands of dollars, for public school teachers for the year t can be modeled by the equation y = 33.374 + 9.162 In t, where t is the number of years since 2000 (i.e, t=4 represents 2004 and so on). 23. According to the model, what was the average salary for public school teachers in 2017? 24. According to the model, dùring which year did the average salary for public school teachers reach $50,000? 25 Sunnosaa counle has $125.000 in a retirement account which navs 6% comnounded continuouslv. D Focus O 55°F Clou lyp to 144 dele 8. 6. backs R T J K N prt scarrow_forward
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