lecture_25

lecture_25 - MA 36600 LECTURE NOTES: MONDAY, MARCH 23...

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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 { f 1 , f 2 , ..., f n } . We say that this is a linearly independent set if the only solution to the equation k 1 f 1 ( t ) + k 2 f 2 ( t ) + + k n f n ( t ) = 0 for all t is k 1 = k 2 = = k n = 0. We say that this is a linearly dependent set otherwise. We explain the relationship with the Wronskian , as the n n determinant, W ( f 1 , f 2 , ..., f n ) ( t ) = f 1 ( t ) f 2 ( t ) f n ( t ) f (1) 1 ( t ) f (1) 2 ( t ) f (1) n ( t ) . . . . . . . . . . . . f ( n- 1) 1 ( t ) f ( n- 1) 2 ( t ) f ( n- 1) n ( t ) . We will show that the following are equivalent: i. W ( f 1 , f 2 , ..., f n ) ( t ) 6 = 0 for some t . ii. W ( f 1 , f 2 , ..., f n ) ( t ) 6 = 0 for all t . iii. { f 1 , f 2 , ..., f n } is a linearly independent set. We give the proof, following Lecture #16 on Monday, February 23. Consider the following propositions: p 1 = W ( f 1 , f 2 , ..., f n ) ( t ) 6 = 0 for some t p 2 = W ( f 1 , f 2 , ..., f n ) ( t ) 6 = 0 for all t p 3 = { f 1 , f 2 , ..., f n } is a linearly independent set We have already seen that p 1 p 2 ; this is Abels Theorem: Say that { y 1 , y 2 , ..., y n } is a set of functions satisfying the equation P ( t ) d n y dt n + P 1 ( t ) d n- 1 y dt...
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lecture_25 - MA 36600 LECTURE NOTES: MONDAY, MARCH 23...

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