527solns5 - 642:527 SOLUTIONS ASSIGNMENT 5 FALL 2009 Some of these solutions were written by Professor Dan Ocone 5.5.1(a(d See solutions in text(b

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642:527 SOLUTIONS: ASSIGNMENT 5 FALL 2009 Some of these solutions were written by Professor Dan Ocone. 5.5.1 (a), (d): See solutions in text. (b) f ( t ) = H ( t ) e t - H ( t - 1) e t . The frst H ( t ) is in a sense not needed, since in discussing the Laplace trans±orm we always take t 0. To take the Laplace trans±orm using (9a) we write f ( t ) = H ( t ) e t - e 1 H ( t - 1) e ( t 1) and fnd F ( s ) = (1 - e 1 s ) / ( s + 1). (c) f ( t ) = 2 H ( t ) - 5 H ( t - 5) + 3 H ( t - 7), F ( s ) = (2 - 5 e 5 s + 3 e 7 s ) /s . 5 (d) See solution in text. 7 (d) The equation is x ′′ - x = 10( H ( t - 5) - H ( t - 7)), so ( s 2 - 1) X ( s ) = 10( e 5 s - e 7 s ) /s and X ( s ) = 10( e 5 s - e 7 s ) p 1 s ( s 2 - 1) P = 10( e 5 s - e 7 s ) p 1 2( s - 1) - 1 s + 1 2( s + 1) P . We take the inverse Laplace trans±orm with the use o± Appendix C, ±ormulas 1, 2, and 30, to obtain x ( t ) = 5 H ( t - 5)( e t 5 - 2 + e ( t 5) ) - 5 H ( t - 7)( e t 7 - 2 + e ( t 7) ) 5.6.1 (a) See solution in text.
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This note was uploaded on 12/15/2009 for the course MATH 527 taught by Professor Staff during the Fall '08 term at Rutgers.

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527solns5 - 642:527 SOLUTIONS ASSIGNMENT 5 FALL 2009 Some of these solutions were written by Professor Dan Ocone 5.5.1(a(d See solutions in text(b

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