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section%208.1

section%208.1 - Chapter 8 Systems of Linear First-Order...

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Chapter 8. Systems of Linear First-Order Di ff erential Equations § 8.1 preliminary Theory 1. Linear Systems: dx 1 dt = a 11 ( t ) x 1 + a 12 ( t ) x 2 + · · · + a 1 n ( t ) x n + f 1 ( t ) dx 2 dt = a 21 ( t ) x 1 + a 22 ( t ) x 2 + · · · + a 2 n ( t ) x n + f 2 ( t ) . . . . . . dx n dt = a n 1 ( t ) x 1 + a n 2 ( t ) x 2 + · · · + a nn ( t ) x n + f n ( t ) Matrix form: Let X = x 1 ( t ) x 2 ( t ) . . . x n ( t ) , A = a 11 ( t ) a 12 ( t ) · · · a 1 n ( t ) a 21 ( t ) a 22 ( t ) · · · a 2 n ( t ) . . . . . . a n 1 ( t ) a n 2 ( t ) · · · a nn ( t ) , F = f 1 ( t ) f 2 ( t ) . . . f n ( t ) Then we have the matrix form of the system X = AX + F . And the initial value problem can be written as X = AX + F , X ( t 0 ) = X 0 2. Some Results. Thm (Existence of a Unique Solution): Let the entries of the matrices A and F be functions continuous on a common interval I that contains t 0 . Then there exists a unique solution of the initial value

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