lecture_31 (dragged)

lecture_31 (dragged) - MA 36600 LECTURE NOTES: FRIDAY,...

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MA 36600 LECTURE NOTES: FRIDAY, APRIL 10 Systems of Linear Equations Systems of Linear Di±erential Equations. We return to the linear system x ° 1 = p 11 ( t ) x 1 + p 12 ( t ) x 2 + ··· + p 1 n ( t ) x n + g 1 ( t ) x ° 2 = p 21 ( t ) x 1 + p 22 ( t ) x 2 + ··· + p 2 n ( t ) x n + g 2 ( t ) . . . . . . x ° m = p m 1 ( t ) x 1 + p m 2 ( t ) x 2 + ··· + p mn ( t ) x n + g m ( t ) Using the notation of matrices, we may express this in the form d dt x 1 x 2 . . . x m ° ±² ³ m -dim’l vector = p 11 ( t ) p 12 ( t ) ··· p 1 n ( t ) p 21 ( t ) p 22 ( t ) ··· p 2 n ( t ) . . . . . . . . . . . . p m 1 ( t ) p m 2 ( t ) ··· p mn ( t ) ° ±² ³ m × n matrix x 1 x 2 . . . x n ° ±² ³ n -dim’l vector + g 1 ( t ) g 2 ( t ) . . . g m ( t ) ° ±² ³ m -dim’l vector When m = n , we can express this in the rather compact form d dt x = P ( t ) x + g ( t ) . We are motivated by the following observation: When m = n = 1, we have the Frst order equation x ° = p ( t ) x + g ( t ) which we can solve using integrating factors: μ ( t )=exp
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