Differential Equations: An Introduction to Modern Methods and Applications
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 15P

Undetermined Coefficients. For each of the nonhomogeneous terms specified in Problems 14 through 16, use the method of undetermined coefficients to find a particular solution of

x ' = ( 2 1 1 2 ) x + g ( t ) = Ax + g ( t ) (ii)

Given that the general solution of the corresponding homogeneous system x' = Ax is

x = c 1 e 3 t ( 1 1 ) + c 2 e t ( 1 1 )

g ( t ) = ( sin t 0 ) = sin t ( 1 0 ) .

Since the entry of g ( t ) contains a sine function, substitute an expression of the form

x p ( t ) = ( cos t ) a + ( sin t ) b = cos t ( a 1 a 2 ) + sin t ( b 1 b 2 )

Into Eq. (ii) and match the coefficients of the sine function and the cosine function on both sides of the resulting equation to obtain the two systems

Aa = b , Ab = a ( 1 0 ) .

Show that ( A 2 +I 2 ) a = ( 1 0 ) and solvefor a . Then substitute thisresult into the second equation and solve for b .

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Chapter 6 Solutions

Differential Equations: An Introduction to Modern Methods and Applications

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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