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

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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 nth order homogeneous linear differential equation dn y dn−1 y dy P0 (t) n + P1 (t) n−1 + · · · + Pn−1 (t) + Pn (t) y = 0. dt dt dt We may turn this into a system of equations as follows. Make the substitution y (1) y y (1) y (2) y (1) y (2) . . d . . . = = P(t) x . . x= =⇒ x= . . dt (n−1) y y (n−2) (n−1) y n−1 ￿ Pn−j (t) (j ) − y y (n−1) y ( n) P0 (t) j =0 in terms of the n × n matrix P(t) = 0 0 1 . . . . . . 0 0 Pn (t) P0 (t) − Pn−1 (t) P0 (t) − ··· 0 .. . . . . ··· 0 0 ··· Pn−2 (t) P0 (t) ··· 0 − . . . . 1 P1 (t) − P0 (t) 0 P2 (t) P0 (t) is a set of functions satisfying the nth order homogeneous equation. We may (n−1) yk 0 0 Now say that {y1 , y2 , . . . , yn } define the vectors yk y (1) k (k ) x = . . . 1 . . . − 0 =⇒ d (k ) x = P(t) x(k) dt The Wronskian is the n × n determinant ￿ ￿ x11 (t) x12 (t) · · · ￿ ￿ ￿ x21 (t) x22 (t) · · · ￿ (1) ￿ ￿ ( n) W x , ..., x (t) = ￿ . . .. ￿ . . . . . ￿ ￿ ￿ x (t) x (t) · · · n1 n2 ￿ ￿ = W y1 , y2 , . . . , yn (t). 1 for k = 1, 2, . . . , n. ￿￿ x1n (t) ￿ ￿ y1 ￿￿ ￿ ￿ (1) x2n (t) ￿ ￿ y1 ￿￿ ￿=￿ . . ￿￿. . . ￿￿. ￿￿ ￿ xnn (t) ￿ ￿y (n−1) 1 y2 (1) y2 . . . (n−1) y2 ··· ··· .. . ··· ￿ yn ￿ ￿ (1) ￿ yn ￿ ￿ ￿ .￿ .￿ .￿ ￿ (n−1) ￿ yn ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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