Explanation Approach: The procedure to solve the general solution to the system x ′ ( t ) = A x ( t ) + f ( t ) . Step-1: Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues: Step-2: Apply the | A − λ I | X = 0 in order to calculate eigenvectors u 1 and u 2 corresponding to eigenvalue λ 1 and λ 2 respectively. Step-3: The complementary solution of the system x ′ ( t ) = A x ( t ) + f ( t ) : x c ( t ) = c 1 e λ 1 t u 1 + c 2 e λ 2 t u 2 Step-4: Assume the particular solution the system x ′ ( t ) = A x ( t ) + f ( t ) as x p ( t ) = [ f ( t ) g ( t ) ] Step-5: Satisfied the system x ′ ( t ) = A x ( t ) + f ( t ) : Step-6: Solve the linear equation and then find: x p ( t ) = [ f ( t ) g ( t ) ] The general solution of system x ′ ( t ) = A x ( t ) + f ( t ) : x ( t ) = x p ( t ) + x c ( t ) Calculation: Consider the general solution to the system x ′ ( t ) = A x ( t ) + f ( t ) A = [ 2 2 2 2 ] , f ( t ) = [ − 4 cos t − sin t ] Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues | [ 2 2 2 2 ] − λ [ 1 0 0 1 ] | = 0 | 2 − λ 2 2 2 − λ | = 0 Expand the above determinant: ( 2 − λ ) ( 2 − λ ) − 4 = 0 ( 2 − λ ) 2 − 2 2 = 0 λ ( λ − 4 ) = 0 Equate the each product equal to zero: λ = 0 and λ = 4 Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = 0 : [ 2 − 0 2 2 2 − 0 ] [ x 1 x 2 ] = [ 0 0 ] [ 2 2 2 2 ] [ x 1 x 2 ] = [ 0 0 ] Write above matrix notation in scalar form notation: 2 x 1 + 2 x 2 = 0 2 x 1 + 2 x 2 = 0 Simplify further, x 1 = − x 2 Let assume x 1 = s and substitute in the above equation, x 1 = s x 2 = − s The eigenvector corresponding to eigenvalue λ = 0 : v 1 = [ s − s ] = s [ 1 − 1 ] The eigenvector corresponding to eigenvalue λ = 0 : v 1 = [ 1 − 1 ] Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = 4 : [ 2 − 4 2 2 2 − 4 ] [ x 1 x 2 ] = [ 0 0 ] [ − 2 2 2 − 2 ] [ x 1 x 2 ] = [ 0 0 ] Write above matrix notation in scalar form notation:

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 9.7, Problem 4E

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To solve:

The general solution to the system x′(t)=Ax(t)+f(t) with help of the method of undetermined coefficient where A and f(t) are given.

A=[2222], f(t)=[−4cost−sint]

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Chapter 9.7, Problem 3E

Chapter 9.7, Problem 5E

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Chapter 9 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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The general solution to the system x ′ ( t ) = A x ( t ) + f ( t ) with help of the method of undetermined coefficient where A and f ( t ) are given. A = [ 2 2 2 2 ] , f ( t ) = [ − 4 cos t − sin t ]

Solution Summary: The author explains the procedure to solve the general solution to the system xprime

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To solve:

The general solution to the system x′(t)=Ax(t)+f(t) with help of the method of undetermined coefficient where A and f(t) are given.

A=[2222], f(t)=[−4cost−sint]

Want to see the full answer?

Chapter 9 Solutions

The general solution to the system x ′ ( t ) = A x ( t ) + f ( t ) with help of the method of undetermined coefficient where A and f ( t ) are given. A = [ 2 2 2 2 ] , f ( t ) = [ − 4 cos t − sin t ]

Solution Summary: The author explains the procedure to solve the general solution to the system xprime

Videos

To solve: The general solution to the system x′(t)=Ax(t)+f(t) with help of the method of undetermined coefficient where A and f(t) are given. A=[2222], f(t)=[−4cost−sint]

Want to see the full answer?

Chapter 9 Solutions

To solve:

The general solution to the system x′(t)=Ax(t)+f(t) with help of the method of undetermined coefficient where A and f(t) are given.

A=[2222], f(t)=[−4cost−sint]