Chem Differential Eq HW Solutions Fall 2011 140

Chem Differential Eq HW Solutions Fall 2011 140 - 140...

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Unformatted text preview: 140 Chapter 8 The Laplace and Hankel Transforms with Applications Solutions to Exercises 8.2 1. To compute the Laplace transform of f (t) = U0(t − 1) − t + 1, use L [ U 0 (t − a)] (s) = e−as ; s so L [ U 0 (t − 1) − t + 1] (s) = L [ U 0 (t − 1)] − L [t] + L [1] = 1 e−s 1 − 2+ . s s s 5. Use the identity sin t = − sin(t − π). Then L [sin t U0(t − π)] (s) = −L [sin(t − π) U0 (t − π)] (s) = −e−πs L [sin t] (s) = −e−πs . s2 + 1 9. y = 2 ( U 0 (t − 2) − U 0 (t − 3)) ; Y = 2 e−2s e−3s − s s 13. y ( U 0 (t − 1) − U 0 (t − 4)) + (t − 5) ( U 0 (t − 4) − U 0(t − 5)) = Y = U 0 (t − 1) − U 0 (t − 4) + (t − 5) U 0 (t − 5) + (t − 4) U 0 (t − 4) − U 0 (t − 4); = e−s e−4s e−4s e−5s −2 + 2− 2 s s s s The following is a variation on Exercise 13. 13 bis. Find the Laplace transform of the function in the picture ...
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