Explanation Given: The given system is X ′ = ( − 1 1 4 1 − 1 ) X . And the given value of the solution is X = ( − 1 2 ) e − 3 t 2 . Verification: The given homogenous linear system can be written in the matrix form as, X ′ = ( − 1 1 4 1 − 1 ) X (1) The value of x and y is obtained as, X = ( − 1 2 ) e − 3 t 2 ( x y ) = ( − e − 3 t 2 2 e − 3 t 2 ) x = − e − 3 t 2 y = 2 e − 3 t 2 Differentiate the expression of x obtained above with respect to t . d x d t = − ( − 3 2 ) e − 3 t 2 (2) Differentiate the expression of y obtained above with respect to t . d y d t = 2 ( − 3 2 ) e − 3 t 2 (3) The value of X ′ is obtained as, X ′ = ( d x d t d y d t ) Substitute the value of d x d t and d y d t from equation (2) and (3) it gives, X ′ = ( − ( − 3 2 ) e − 3 t 2 2 ( − 3 2 ) e − 3 t 2 ) Simplify the above obtained expression

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Author: Dennis G. Zill

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Chapter 10.1, Problem 13E

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To verify: The vector X is a solution of the given homogeneous linear system.

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Chapter 10.1, Problem 12E

Chapter 10.1, Problem 14E

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