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. f(t) = e2t cos 3t

. f(t) = e2t cos 3t

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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THEOREM 1.16 Suppose f is a piecewise ponential order. Then the Specifically, if f(t)| ≤ C EXERCISES In Exercises 1-10, use Definition 1.1 of the Laplace transform to find the Laplace transform of each of the following functions defined for t > 0. 1. f(t) = 3 3. f(t) = e-2t 5. f(t) = cos 2t 7. f(t) = te²t 9. f(t) = e²t cos 3t 11. In the narrative, we used Definition 1.1 to show that the Laplace transform of the function defined by f(t) = t is F(s) = 1/s². Theorem 1.16 says th tions. However, there are For example, the functio shown that it does not hav 2. f(t) = -2 4. f(t) = e³t 6. f(t) = sin 3t 8. f(t) = te-3t 10. f(t) = e-³t sin 2t (a) Use Definition 1.1 to show that the Laplace transform of the function defined by g(t) = t² is G(s) = 2!/s³. (b) Use Definition 1.1 to show that the Laplace transform of the function defined by h(t) = t³ is H(s) = 3!/s4. (c) Use Definition 1.1 and mathematical induction to prove that the Laplace transform of the function de- fined by f(t) = t" is F(s) F(s) = F(s) = = 12. Use Definition 1.1 to show that the Laplace transform of the function defined by f(t) = cos wt is n! tn+1 S s² + w² 13. Use Definition 1.1 to show that the Laplace transform of the function defined by f(t) = eat cos wt is s-a (s-a)² +w² 14. Use Definition 1.1 to show that the Laplace transform of the function defined by f(t) = et sin at is
Transcribed Image Text:THEOREM 1.16 Suppose f is a piecewise ponential order. Then the Specifically, if f(t)| ≤ C EXERCISES In Exercises 1-10, use Definition 1.1 of the Laplace transform to find the Laplace transform of each of the following functions defined for t > 0. 1. f(t) = 3 3. f(t) = e-2t 5. f(t) = cos 2t 7. f(t) = te²t 9. f(t) = e²t cos 3t 11. In the narrative, we used Definition 1.1 to show that the Laplace transform of the function defined by f(t) = t is F(s) = 1/s². Theorem 1.16 says th tions. However, there are For example, the functio shown that it does not hav 2. f(t) = -2 4. f(t) = e³t 6. f(t) = sin 3t 8. f(t) = te-3t 10. f(t) = e-³t sin 2t (a) Use Definition 1.1 to show that the Laplace transform of the function defined by g(t) = t² is G(s) = 2!/s³. (b) Use Definition 1.1 to show that the Laplace transform of the function defined by h(t) = t³ is H(s) = 3!/s4. (c) Use Definition 1.1 and mathematical induction to prove that the Laplace transform of the function de- fined by f(t) = t" is F(s) F(s) = F(s) = = 12. Use Definition 1.1 to show that the Laplace transform of the function defined by f(t) = cos wt is n! tn+1 S s² + w² 13. Use Definition 1.1 to show that the Laplace transform of the function defined by f(t) = eat cos wt is s-a (s-a)² +w² 14. Use Definition 1.1 to show that the Laplace transform of the function defined by f(t) = et sin at is
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