lecture_31

# lecture_31 - MA 36600 LECTURE NOTES FRIDAY APRIL 10 Systems...

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Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, APRIL 10 Systems of Linear Equations Systems of Linear Differential 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 | {z } 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 ) | {z } m × n matrix x 1 x 2 . . . x n | {z } n-dim’l vector + g 1 ( t ) g 2 ( t ) . . . g m ( t ) | {z } 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 first order equation x = p ( t ) x + g ( t ) which we can solve using integrating factors: μ ( t ) = exp- Z t p ( τ ) dτ = ⇒ x ( t ) = 1 μ ( t ) Z t μ ( τ ) g ( τ ) dτ + C ....
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lecture_31 - MA 36600 LECTURE NOTES FRIDAY APRIL 10 Systems...

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