Laplace Tansformation
Q-A. Find the Laplace transform of the following functions 1. f (t) = t − 1, 0 < t < 3; 7, t > 3. 2. f (t) = cos t − 0,
2π 3
, 0 2π . 3
2π ; 3
4, 0 < t < 1; −2, 1 < t < 3; 3. f (t) = 5, t > 3. 5. f (t) = 3t3 + e−2t + t 3 7. f (t) = cos3 2t 9. f (t) = sin (3t + 5) 11. f (t) = e−3t sin2 t 13. f (t) = 7T 15. f (t) = e−3t (cos (4t) + 3 sin (4t)) 17. f (t) = teat 19. f (t) = t sin2 3t 21. f (t) = t2 e−2t cos t 23. f (t) = t cos (7t + 9) 25. f (t) = 27. f (t) = sin2 t t e−t sin t t
1 2 2
4
4. f (t) = 6. f (t) =
sin t, 0 < t < π; 0, t > π. eat − 1 a
8. f (t) = cosh3 3t 10. f (t) = t2 sin (at) 12. f (t) = e−t cos2 t 14. f (t) = cat+b 16. f (t) = 4e−2t cosh (3t) 18. f (t) = t2 cos (3t) 20. f (t) = teat sin (at) 22. f (t) = t2 eat sin (105t) 24. f (t) = t2 e−3t cos (2t) 26. f (t) = sin t t
1 2
28. f (t) = et u t − u t−
1 2
29. f (t) = t −
30. f (t) = cos t u(t − π) 32. f (t) = (1 + 2t + 3t2 − 4t3 ) u(t − 2)
31. f (t) = t2 u(t − 3)
33. f (t) = t − t u(t − π) + sin t u(t − 2π) 34. f (t) = 2t u(t − 2)
Q-B. Find the Laplace inverse transform of the following functions 1. F (s) = 3. F (s) = 5. F (s) = 7. F (s) = s s4 + 4a4 s 2+s+1 s 4s + 12 s2 + 8s + 16 5s − 10 9s2 − 16 s+1 s−1 2. F (s) = 4. F (s) = 6. F (s) = 8. F (s) = s+2 s2 − 4s + 13 (s2 − 1)2 s5 2s − 1 s4 + s2 + 1 1 16 + 4s 1 − s2 1+s s
9. F (s) = log
10. F (s) = log 12. F (s) = 14. F (s) = 16. F (s) =
11. F (s) = cot−1 (s + 1) 13. F (s) = 15. F (s) = 17. F (s) = 19. F (s) = 21. F (s) = 23. F (s) = 25. F (s) = 1 − 2s3
8 s4 − 4s2 s2 4 + 4s 1 + a2 )
s4
1 s(s + a)3 e−2s s−5 se−2s s2 + π 2 1 + e−sπ/2 s2 + 4 se−as s2 + b 2 e−s (s − 1)(s − 2)
s(s2
e−s 18. F (s) = √ s+1 20. F (s) = 22. F (s) = 24. F (s) = 26. F (s) = se−2s s2 + 16 e−πs s+a e−s (s + 1)3 e−4s(s+2) s2 + 4s + 5
Q-C. Using convolution theorem find the Laplace inverse transform of the following functions 1. F (s) = 3. F (s) = 5. F (s) = 7. F (s) = 1 (s + 7)(s + 8) (s2 1 + a2 )2 2. F (s) = 4. F (s)
2π 3
, 0 2π . 3
2π ; 3
4, 0 < t < 1; −2, 1 < t < 3; 3. f (t) = 5, t > 3. 5. f (t) = 3t3 + e−2t + t 3 7. f (t) = cos3 2t 9. f (t) = sin (3t + 5) 11. f (t) = e−3t sin2 t 13. f (t) = 7T 15. f (t) = e−3t (cos (4t) + 3 sin (4t)) 17. f (t) = teat 19. f (t) = t sin2 3t 21. f (t) = t2 e−2t cos t 23. f (t) = t cos (7t + 9) 25. f (t) = 27. f (t) = sin2 t t e−t sin t t
1 2 2
4
4. f (t) = 6. f (t) =
sin t, 0 < t < π; 0, t > π. eat − 1 a
8. f (t) = cosh3 3t 10. f (t) = t2 sin (at) 12. f (t) = e−t cos2 t 14. f (t) = cat+b 16. f (t) = 4e−2t cosh (3t) 18. f (t) = t2 cos (3t) 20. f (t) = teat sin (at) 22. f (t) = t2 eat sin (105t) 24. f (t) = t2 e−3t cos (2t) 26. f (t) = sin t t
1 2
28. f (t) = et u t − u t−
1 2
29. f (t) = t −
30. f (t) = cos t u(t − π) 32. f (t) = (1 + 2t + 3t2 − 4t3 ) u(t − 2)
31. f (t) = t2 u(t − 3)
33. f (t) = t − t u(t − π) + sin t u(t − 2π) 34. f (t) = 2t u(t − 2)
Q-B. Find the Laplace inverse transform of the following functions 1. F (s) = 3. F (s) = 5. F (s) = 7. F (s) = s s4 + 4a4 s 2+s+1 s 4s + 12 s2 + 8s + 16 5s − 10 9s2 − 16 s+1 s−1 2. F (s) = 4. F (s) = 6. F (s) = 8. F (s) = s+2 s2 − 4s + 13 (s2 − 1)2 s5 2s − 1 s4 + s2 + 1 1 16 + 4s 1 − s2 1+s s
9. F (s) = log
10. F (s) = log 12. F (s) = 14. F (s) = 16. F (s) =
11. F (s) = cot−1 (s + 1) 13. F (s) = 15. F (s) = 17. F (s) = 19. F (s) = 21. F (s) = 23. F (s) = 25. F (s) = 1 − 2s3
8 s4 − 4s2 s2 4 + 4s 1 + a2 )
s4
1 s(s + a)3 e−2s s−5 se−2s s2 + π 2 1 + e−sπ/2 s2 + 4 se−as s2 + b 2 e−s (s − 1)(s − 2)
s(s2
e−s 18. F (s) = √ s+1 20. F (s) = 22. F (s) = 24. F (s) = 26. F (s) = se−2s s2 + 16 e−πs s+a e−s (s + 1)3 e−4s(s+2) s2 + 4s + 5
Q-C. Using convolution theorem find the Laplace inverse transform of the following functions 1. F (s) = 3. F (s) = 5. F (s) = 7. F (s) = 1 (s + 7)(s + 8) (s2 1 + a2 )2 2. F (s) = 4. F (s)