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lecture_33 (dragged) - MA 36600 LECTURE NOTES WEDNESDAY...

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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 di ff erential equation P 0 ( 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 j =0 P n j ( t ) P 0 ( t ) y ( j ) = P ( t ) x in terms of the n × n matrix P ( t ) = 0 1 0 · · · 0 0 0 0 1 · · · 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 · · · 0 1 P n ( t ) P 0 ( t ) P n 1 ( t ) P 0 ( t ) P n 2 ( t ) P 0 ( t ) · · · P 2 ( t ) P 0 ( t ) P 1 ( t ) P 0 ( t ) . Now say that { y 1 , y 2 , . . . , y n } is a set of functions satisfying the
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