527solns5

527solns5 - 642:527 SOLUTIONS ASSIGNMENT 5 FALL 2007 Some...

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Unformatted text preview: 642:527 SOLUTIONS: ASSIGNMENT 5 FALL 2007 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 first H ( t ) is in a sense not needed, since in discussing the Laplace transform we always take t ≥ 0. To take the Laplace transform using (9a) we write f ( t ) = H ( t ) e − t- e − 1 H ( t- 1) e − ( t − 1) and find 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 . 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 ) parenleftbigg 1 s ( s 2- 1) parenrightbigg = 10( e − 5 s- e − 7 s ) parenleftbigg 1 2( s- 1)- 1 s + 1 2( s + 1) parenrightbigg . We take the inverse Laplace transform with the use of Appendix C, formulas 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 −...
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527solns5 - 642:527 SOLUTIONS ASSIGNMENT 5 FALL 2007 Some...

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