M334Lec33 - Math 334 Lecture #33 7.7: Fundamental Matrices...

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Unformatted text preview: Math 334 Lecture #33 7.7: Fundamental Matrices Example. Recall that a general solution of ~x = A~x where A = 1 1 4 1 is ~x ( t ) = c 1 ~x (1) + c 2 ~x (2) = c 1 e 3 t 2 e 3 t + c 2 e- t- 2 e- t = e 3 t e- t 2 e 3 t- 2 e- t c 1 c 2 . The form of this general solution is that of an invertible (why?) 2 2 matrix function times a vector of arbitrary constants, ~x ( t ) = ( t ) ~ c, where the columns of the 2 2 the matrix function, ( t ) = e 3 t e- t 2 e 3 t- 2 e- t , are the linearly independent solutions ~x (1) and ~x (2) of the system. The function ( t ) is called a fundamental matrix for the system; it satisfies the first order differential matrix equation ( t ) = A ( t ) where A = 1 1 4 1 . This is true because each column of ( t ) is a solution of the system. This can be verified: ( t ) = 3 e 3 t- e- t 6 e 3 t 2 e- t and 1 1 4 1 ( t ) = 1 1 4 1 e 3 t e- t 2 e 3 t- 2 e- t = 3 e 3 t- e- t 6 e 3 t 2 e- t . X The Form of General Solutions. If ~x (1) ( t ) , . . . , ~x ( n ) ( t ) are n linearly independent solutions of ~x...
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

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M334Lec33 - Math 334 Lecture #33 7.7: Fundamental Matrices...

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