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2 MA 36600 LECTURE NOTES: MONDAY, APRIL 20 Fundamental Matrices Recap. Consider the initial value problem d dt x = P ( t ) x , x ( t 0 )= x 0 . Assuming that P ( t )= ° p ij ( t ) ± is an n × n matrix, say that we can fnd n Functions x (1) , x (2) ,..., x ( n ) such that i. Each x ( k ) ( t ) is a solution to the system d dt x ( k ) = P ( t ) x ( k ) ,k =1 , 2 ,...,n . ii. Their Wronskian is nonzero: W ° x (1) , x (2) ,..., x ( n ) ± ( t )=det x 11 ( tx 12 ( t ) ··· x 1 n ( t ) x 21 ( t ) x 22 ( t ) ··· x 2 n ( t ) . . . . . . . . . . . . x n 1 ( t ) x n 2 ( t ) ··· x nn ( t ) where x ( k ) ( t )= x 1 k ( t ) x 2 k ( t ) . . . x nk ( t ) . Then the general solution oF the system oF di±erential equations is in the Form x ( t )= n ² k =1 c k x ( k ) ( t )= Ψ ( t ) c where Ψ ( t )= x 11 ( tx 12 ( t ) ··· x 1 n ( t ) x 21 ( t ) x 22 ( t ) ··· x 2 n ( t ) . . . . . . . . . . . . x n 1 ( t ) x n 2 ( t ) ··· x nn ( t ) , c = c 1 c 2 . . . c n . We call ³ x (1) , x (2) ,..., x ( n ) ´ a fundamental set of solutions , and Ψ ( t )= ° x ij ( t ) ± a fundamental matrix . Note that the Fundamental matrix satisfes the matrix equation d
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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