lecture_33 - MA 36600 LECTURE NOTES: WEDNESDAY, APRIL 15...

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Unformatted text preview: MA 36600 LECTURE NOTES: WEDNESDAY, APRIL 15 Basic Theory of Systems of First Order Linear Equations Example. We explain how the Wronskian defined in the previous lecture is related to the Wronskian we defined during Lecture #24 on Friday, March 13. Consider the n th order homogeneous linear differential equation P ( t ) d n y dt n + P 1 ( t ) d n- 1 y dt n- 1 + + P n- 1 ( t ) dy dt + P n ( t ) y = 0 . We may turn this into a system of equations as follows. Make the substitution x = y y (1) . . . y ( n- 2) y ( n- 1) = d dt x = y (1) y (2) . . . y ( n- 1) y ( n ) = y (1) y (2) . . . y ( n- 1)- n- 1 X j =0 P n- j ( t ) P ( t ) y ( j ) = P ( t ) x in terms of the n n matrix P ( t ) = 1 1 . . . . . . . . . . . . . . . . . . 1- P n ( t ) P ( t )- P n- 1 ( t ) P ( t )- P n- 2 ( t ) P ( t ) - P 2 ( t ) P ( t )- P 1 ( t ) P ( t ) . Now say that { y 1 , y 2 , ..., y n } is a set of functions satisfying the n th order homogeneous equation. We may define the vectors x ( k ) = y k y (1) k ....
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This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue University-West Lafayette.

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lecture_33 - MA 36600 LECTURE NOTES: WEDNESDAY, APRIL 15...

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