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lecture_33 (dragged) 1 - 2 MA 36600 LECTURE NOTES WEDNESDAY...

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2 MA 36600 LECTURE NOTES: WEDNESDAY, APRIL 15 Hence the defnitions are consistent. As For Abel’s Theorem, we have the trace tr P ( t )= P 1 ( t ) P 0 ( t ) = W ( t )= C exp ° tr ± t P ( τ ) ² = C exp ° ± t P 1 ( τ ) P 0 ( τ ) ² . Homogeneous Systems. Again, consider the homogeneous system d dt x = P ( t ) x . Say ³ x (1) ( t ) ,..., x ( n ) ( t ) ´ is a set oF solutions, and denote the n × n matrix Ψ ( t )= x 11 ( t ) x 12 ( t ) ··· x 1 n ( t ) x 21 ( t ) x 22 ( t ) ··· x 2 n ( t ) . . . . . . . . . . . . x n 1 ( t ) x n 2 ( t ) ··· x nn ( t ) where x ( k ) ( t )= x 1 k ( t ) x 2 k ( t ) . . . x nk ( t ) . The Following are equivalent: i. ³ x (1) ( t ) ,..., x ( n ) ( t ) ´ is a Fundamental set oF solutions. ii. ³ x (1) ( t ) ,..., x ( n ) ( t ) ´ is a linearly independent set. iii. Ψ ( t )= µ x ij ( t ) is a nonsingular matrix For all t . iv. Ψ ( t 0 )= µ x ij ( t 0 ) is a nonsingular matrix For some t 0 . v. W µ x (1) ,..., x ( n ) ( t ) ° = 0 For all t . vi. W µ x (1) ,..., x ( n ) ( t 0 ) ° = 0 For some t 0 . We sketch the prooF Following the diagram below: (i) ° (iii) ± ² (v) ³ ´ (ii) ° (iv) ° (vi) µ By defnition, (i) ⇐⇒ (iii). Since
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