Explanation Given: X p is a particular solution to the nonhomogeneous system X ′ = A ( t ) X ( t ) + f ( t ) and { X 1 , ⋯ , X n } is a fundamental solution set on I for the corresponding homogeneous system X ′ = A ( t ) X ( t ) . Approach: Theorem: If X p is a particular solution to the nonhomogeneous system X ′ = A ( t ) X ( t ) + f ( t ) on the interval I and { X 1 , ⋯ , X n } is a fundamental solution set on I for the corresponding homogeneous system X = A ( t ) X ( t ) . Then every solution to above equation on I can be expressed in the form X ( t ) = X p ( t ) + c 1 X 1 ( t ) + ⋯ + c n X n ( t ) , where c 1 , ⋯ , c n are constant. Proof: Consider the expression, X ( t ) = X p ( t ) + c 1 X 1 ( t ) + ⋯ + c n X n ( t ) . Substitute in the system X ′ = A ( t ) X ( t ) + f ( t ) : A ( t ) X ( t ) + f ( t ) = A ( t ) { X p ( t ) + c 1 X 1 ( t ) + ⋯ + c n X n ( t ) } + f ( t ) = A ( t ) X p ( t ) + A ( t ) { c 1 X 1 ( t ) + ⋯ + c n X n ( t ) } + f ( t )

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

7th Edition

ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

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Chapter 9.4, Problem 36E

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To prove:

The given solution of the system X′=A(t)X(t)+f(t) on I can be expressed in the form of X(t)=Xp(t)+c1X1(t)+⋯+cnXn(t).

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Chapter 9.4, Problem 35E

Chapter 9.4, Problem 37E

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Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

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To prove: The given solution of the system X ′ = A ( t ) X ( t ) + f ( t ) on I can be expressed in the form of X ( t ) = X p ( t ) + c 1 X 1 ( t ) + ⋯ + c n X n ( t ) .

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To prove: The given solution of the system X′=A(t)X(t)+f(t) on I can be expressed in the form of X(t)=Xp(t)+c1X1(t)+⋯+cnXn(t).

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Chapter 9 Solutions

To prove:

The given solution of the system X′=A(t)X(t)+f(t) on I can be expressed in the form of X(t)=Xp(t)+c1X1(t)+⋯+cnXn(t).