Explanation Given: The given X ( t ) is: X ( t ) = [ e − t e 5 t − e 2 t e 5 t ] The given initial value problem is: X ′ = [ 2 3 3 2 ] X , X ( 0 ) = [ 3 − 1 ] Approach: The matrix X ( t ) is a fundamental matrix if it satisfy the system X ′ = A X . The inverse of the matrix can be calculate by reducing: [ X ( t ) : I ] → [ I : X ( t ) ] The solution of initial of initial value problem is: x ( t ) = X ( t ) X − 1 ( 0 ) X 0 Calculation: Consider the matrix: X ( t ) = [ e − t e 5 t − e 2 t e 5 t ] Differentiate the with respect to t . X ′ ( t ) = [ ( e − t ) ′ ( e 5 t ) ′ ( − e 2 t ) ′ ( e 5 t ) ′ ] = [ − e − t 5 e 5 t − 2 e 2 t 5 e 5 t ] Consider the system and substitute the value of X ( t ) : [ 2 3 3 2 ] X = [ 2 3 3 2 ] [ e − t e 5 t − e 2 t e 5 t ] = [ − e − t 5 e 5 t − 2 e 2 t 5 e 5 t ] Therefore [ − e − t 5 e 5 t − 2 e 2 t 5 e 5 t ] = [ − e − t 5 e 5 t − 2 e 2 t 5 e 5 t ] , the X ( t ) is the fundamental matrix. Compute the inverse of X ( t ) : | X ( t ) | = | e − t e 5 t − e 2 t e 5 t | = e − t ⋅ e 5 t − ( − e 2 t ) ⋅ e 5 t = e 4 t + e 7 t The transpose of co factor matrix is: a 11 ( t ) = e 5 t a 12 ( t ) = e 2 t a 21 ( t ) = − e 5 t a 22 ( t ) = e − t The matrix form is: A c = [ e 5 t − e 5 t e 2 t e − t ] The inverse matrix is:

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

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

Publisher: Addison Wesley

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Chapter 9.4, Problem 30E

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The X(t) is a fundamental matrix for the given system, compute the X−1(t) and find the solution of given initial value problem.

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Chapter 9.4, Problem 29E

Chapter 9.4, Problem 31E

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To verify: The X ( t ) is a fundamental matrix for the given system, compute the X − 1 ( t ) and find the solution of given initial value problem.

Solution Summary: The author explains that the matrix X(t) is a fundamental matrix for the given system.

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To verify:

The X(t) is a fundamental matrix for the given system, compute the X−1(t) and find the solution of given initial value problem.

Want to see the full answer?

Chapter 9 Solutions

To verify: The X ( t ) is a fundamental matrix for the given system, compute the X − 1 ( t ) and find the solution of given initial value problem.

Solution Summary: The author explains that the matrix X(t) is a fundamental matrix for the given system.

Videos

To verify: The X(t) is a fundamental matrix for the given system, compute the X−1(t) and find the solution of given initial value problem.

Want to see the full answer?

Chapter 9 Solutions

To verify:

The X(t) is a fundamental matrix for the given system, compute the X−1(t) and find the solution of given initial value problem.