Explanation Formula used: The procedure to solve the x ′ ( t ) = A x ( 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:Write the general solution of system x ′ ( t ) = A x ( t ) as: X ( t ) = c 1 e λ 1 t u 1 + c 2 e λ 2 t u 2 Calculation: Consider the system x ′ ( t ) = A x ( t ) , A = [ 5 0 0 0 − 4 3 0 3 4 ] Apply the condition | A − λ I | = 0 for matrix A in order to calculate eigenvalues: | [ 5 0 0 0 − 4 3 0 3 4 ] − λ [ 1 0 0 0 1 0 0 0 1 ] | = 0 | 5 − λ 0 0 0 − 4 − λ 3 0 3 4 − λ | = 0 Expand the above determinant: ( 5 − λ ) ( 4 + λ ) ( 4 − λ ) = 0 Equate first factor to 0: ( λ − 4 ) = 0 λ = 4 Equate second factor to 0: ( λ + 4 ) = 0 λ = − 4 Equate third factor to 0: ( λ − 5 ) = 0 λ = 5 Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = 4 : [ 5 − 4 0 0 0 − 4 − 4 3 0 3 4 − 4 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] [ 1 0 0 0 − 8 3 0 3 0 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] Write above matrix notation in scalar form notation: x 1 = 0 − 8 x 2 + 3 x 3 = 0 3 x 2 = 0 The eigenvector corresponding to eigenvalue λ = 4 : u 1 = [ 0 0 0 ] Apply the | A − λ I | X = 0 in order to calculate eigenvector corresponding to eigenvalue λ = − 4 :

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Chapter 9.RP, Problem 6RP

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To calculate: The fundamental matrix for the system x′(t)=Ax(t), where A is given

A=[5000−43034].

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Chapter 9.RP, Problem 5RP

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The fundamental matrix for the system x ′ ( t ) = A x ( t ) , where A is given A = [ 5 0 0 0 − 4 3 0 3 4 ] .

Solution Summary: The author explains the formula used to calculate the fundamental matrix for the system xprime

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To calculate: The fundamental matrix for the system x′(t)=Ax(t), where A is given

A=[5000−43034].

Want to see the full answer?

Chapter 9 Solutions

The fundamental matrix for the system x ′ ( t ) = A x ( t ) , where A is given A = [ 5 0 0 0 − 4 3 0 3 4 ] .

Solution Summary: The author explains the formula used to calculate the fundamental matrix for the system xprime

Videos

To calculate: The fundamental matrix for the system x′(t)=Ax(t), where A is given A=[5000−43034].

Want to see the full answer?

Chapter 9 Solutions

To calculate: The fundamental matrix for the system x′(t)=Ax(t), where A is given

A=[5000−43034].