Chapter 4

# Chapter 4 - 4 1 The Laplace Transform Exercises 4.1 1 1...

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4 The Laplace Transform Exercises 4.1 1. { f ( t ) } = 1 0 e st dt + 1 e st dt = 1 s e st 1 0 1 s e st 1 = 1 s e s 1 s 0 1 s e s = 2 s e s 1 s , s > 0 2. { f ( t ) } = 2 0 4 e st dt = 4 s e st 2 0 = 4 s ( e 2 s 1) , s > 0 3. { f ( t ) } = 1 0 te st dt + 1 e st dt = 1 s te st 1 s 2 e st 1 0 1 s e st 1 = 1 s e s 1 s 2 e s 0 1 s 2 1 s (0 e s ) = 1 s 2 (1 e s ) , s > 0 4. { f ( t ) } = 1 0 (2 t + 1) e st dt = 2 s te st 2 s 2 e st 1 s e st 1 0 = 2 s e s 2 s 2 e s 1 s e s 0 2 s 2 1 s = 1 s (1 3 e s ) + 2 s 2 (1 e s ) , s > 0 5. { f ( t ) } = π 0 (sin t ) e st dt = s s 2 + 1 e st sin t 1 s 2 + 1 e st cos t π 0 = 0 + 1 s 2 + 1 e πs 0 1 s 2 + 1 = 1 s 2 + 1 ( e πs + 1) , s > 0 6. { f ( t ) } = π/ 2 (cos t ) e st dt = s s 2 + 1 e st cos t + 1 s 2 + 1 e st sin t π/ 2 = 0 0 + 1 s 2 + 1 e πs/ 2 = 1 s 2 + 1 e πs/ 2 , s > 0 7. f ( t ) = 0 , 0 < t < 1 t, t > 1 { f ( t ) } = 1 te st dt = 1 s te st 1 s 2 e st 1 = 1 s e s + 1 s 2 e s , s > 0 8. f ( t ) = 0 , 0 < t < 1 2 t 2 , t > 1 { f ( t ) } = 2 1 ( t 1) e st dt = 2 1 s ( t 1) e st 1 s 2 e st 1 = 2 s 2 e s , s > 0 9. f ( t ) = 1 t, 0 < t < 1 0 , t > 0 { f ( t ) } = 1 0 (1 t ) e st dt = 1 s (1 t ) e st + 1 s 2 e st 1 0 = 1 s 2 e s + 1 s 1 s 2 , s > 0 10. f ( t ) = 0 , 0 < t < a c, a < t < b 0 , t > b ; { f ( t ) } = b a ce st dt = c s e st b a = c s ( e sa e sb ), s > 0 165

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Exercises 4.1 11. { f ( t ) } = 0 e t +7 e st dt = e 7 0 e (1 s ) t dt = e 7 1 s e (1 s ) t 0 = 0 e 7 1 s = e 7 s 1 , s > 1 12. { f ( t ) } = 0 e 2 t 5 e st dt = e 5 0 e ( s +2) t dt = e 5 s + 2 e ( s +2) t 0 = e 5 s + 2 , s > 2 13. { f ( t ) } = 0 te 4 t e st dt = 0 te (4 s ) t dt = 1 4 s te (4 s ) t 1 (4 s ) 2 e (4 s ) t 0 = 1 (4 s ) 2 , s > 4 14. { f ( t ) } = 0 t 2 e 2 t e st dt = 0 t 2 e ( s +2) t dt = 1 s + 2 t 2 e ( s +2) t 2 ( s + 2) 2 te ( s +2) t 2 ( s + 2) 3 e ( s +2) t 0 = 2 ( s + 2) 3 , s > 2 15. { f ( t ) } = 0 e t (sin t ) e st dt = 0 (sin t ) e ( s +1) t dt = ( s + 1) ( s + 1) 2 + 1 e ( s +1) t sin t 1 ( s + 1) 2 + 1 e ( s +1) t cos t 0 = 1 ( s + 1) 2 + 1 = 1 s 2 + 2 s + 2 , s > 1 16. { f ( t ) } = 0 e t (cos t ) e st dt = 0 (cos t ) e (1 s ) t dt = 1 s (1 s ) 2 + 1 e (1 s ) t cos t + 1 (1 s ) 2 + 1 e (1 s ) t sin t 0 = 1 s (1 s ) 2 + 1 = s 1 s 2 2 s + 2 , s > 1 17. { f ( t ) } = 0 t (cos t ) e st dt = st s 2 + 1 s 2 1 ( s 2 + 1) 2 (cos t ) e st + t s 2 + 1 + 2 s ( s 2 + 1) 2 (sin t ) e st 0 = s 2 1 ( s 2 + 1) 2 , s > 0 18. { f ( t ) } = 0 t (sin t ) e st dt = t s 2 + 1 2 s ( s 2 + 1) 2 (cos t ) e st st s 2 + 1 + s 2 1 ( s 2 + 1) 2 (sin t ) e st 0 = 2 s ( s 2 + 1) 2 , s > 0 19. { 2 t 4 } = 2 4! s 5 20. { t 5 } = 5! s 6 21. { 4 t 10 } = 4 s 2 10 s 22. { 7 t + 3 } = 7 s 2 + 3 s 23. { t 2 + 6 t 3 } = 2 s 3 + 6 s 2 3 s 24. {− 4 t 2 + 16 t + 9 } = 4 2 s 3 + 16 s 2 + 9 s 25. { t 3 + 3 t 2 + 3 t + 1 } = 3! s 4 + 3 2 s 3 + 3 s 2 + 1 s 26. { 8 t 3 12 t 2 + 6 t 1 } = 8 3!
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• Fall '07
• Delahanty

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