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lecture_35 (dragged) 1 - 2 MA 36600 LECTURE NOTES MONDAY...

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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 find 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 ( t x 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 ff erential equations is in the form x ( t ) = n k =1 c k x ( k ) ( t ) = Ψ ( t ) c where Ψ ( t ) = x 11 ( t x 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 x (1) , x (2) , . . . , x ( n ) a fundamental set of solutions , and Ψ ( t ) = x ij ( t ) a fundamental matrix .
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