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Unformatted text preview: MA 36600 LECTURE NOTES: MONDAY, MARCH 23 Higher Order Linear Equations Linear Independence. Consider a collection of functions {f1 , f2 , . . . , fn }. We say that this is a linearly independent set if the only solution to the equation k1 f1 (t) + k2 f2 (t) + · · · + kn fn (t) = 0 for all t is k1 = k2 = · · · = kn = 0. We say that this is a linearly dependent set with the Wronskian, as the n × n determinant, ￿ ￿ f1 (t) f2 (t) ￿ ￿ (1) (1) ￿ f (t) f2 (t) ￿1 ￿ ￿ W f1 , f2 , . . . , fn (t) = ￿ ￿ . . . . ￿ . . ￿ ￿ (n−1) (n−1) ￿f (t) f2 (t) 1 otherwise. We explain the relationship ￿ fn (t) ￿ ￿ ￿ (1) fn (t) ￿ ￿ ￿. ￿ . . ￿ . ￿ ￿ (n−1) fn (t)￿ ··· ··· .. . ··· We will show that the following are equivalent: ￿ ￿ i. W ￿f1 , f2 , . . . , fn ￿(t0 ) ￿= 0 for some t0 . ii. W f1 , f2 , . . . , fn (t) ￿= 0 for all t. iii. {f1 , f2 , . . . , fn } is a linearly independent set. We give the proof, following Lecture #16 on Monday, February 23. Consider the following propositions: ￿ ￿ p1 = “W f1 , f2 , . . . , fn (t0 ) ￿= 0 for some t0 ” ￿ ￿ p2 = “W f1 , f2 , . . . , fn (t) ￿= 0 for all t” p3 = “{f1 , f2 , . . . , fn } is a linearly independent set” We have already seen that p1 ⇐⇒ p2 ; this is Abel’s Theorem: Say that {y1 , y2 , . . . , yn } is a set of functions satisfying the equation P0 (t) dn−1 y dy dn y + P1 (t) n−1 + · · · + Pn−1 (t) + Pn (t) y = 0. n dt dt dt Then their Wronskian satisfies P0 (t) dW + P1 (t) W = 0 dt =⇒ ￿￿ ￿ ￿ W y1 , y2 , . . . , yn (t) = C exp − t We show that p2 ⇐⇒ p3 , so consider the equation k1 f1 (t) + k2 f2 (t) + · · · + kn fn (t) = 0 for all t. Since this is true for all t, we can differentiate this equation (n − 1) times: f1 (t) k1 (1) f1 (t) k1 (n−1) f1 (t) k1 + f2 (t) k2 + (1) f2 (t) k2 (n−1) + f2 (t) k2 + ··· + fn (t) kn =0 + ··· + (1) fn (t) kn =0 + ··· + fn 1 . . . (n−1) (t) kn =0 ￿ P1 (τ ) dτ . P0 (τ ) ...
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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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