In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.] 1. t² + e¹ sin 2t 2. 31²-²1
In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.] 1. t² + e¹ sin 2t 2. 31²-²1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![In Problems 1-20, determine the Laplace transform of the
given function using Table 7.1 on page 356 and the properties
of the transform given in Table 7.2. [Hint: In Problems 12-20,
use an appropriate trigonometric identity.]
1. t² + e¹ sin 2t
3. e¹cos 3t+e6t1
5. 2t²e¹t+ cos 4t
7. (t-1)4
9. et sin 2t
11. cosh bt
13. sin² t
15. cos³ t
17. sin 2t sin 5t
19. cos nt sin mt, m # n
2. 31²-21
4. 3t2t² + 1
6. e 2¹ sin2t + e³4²
8. (1+e¯¹)²
10. te²¹ cos 5t
12. sin 3t cos 3t
t
14. etsin²
16. t sin² t
18. cosnt cos mt, mn
20. t sin 2t sin 5t
21. Given that L{cos bt} (s) = s/(s² + b²), use the trans-
lation property to compute L{e" cos bt}.
22. Starting with the transform L{1}(s) 1/s, use for-
mula (6) for the derivatives of the Laplace transform
to show that L{t}(s) = 1/s², L{t²} (s) = 2!/s³,
and, by using induction, that L{f} (s) = n!/s"+¹,
n = 1, 2, ....
=
23. Use Theorem 4 on page 362 to show how entry 32 fol-
lows from entry 31 in the Laplace transform table on the
inside back cover of the text.
24. Show that Leªth} (s) = n!/(s-a)"+¹ in two ways:
(a) Use the translation property for F(s).
(b) Use formula (6) for the derivatives of the Laplace
transform.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7a11c6-35df-4daf-855c-043e8f69fb75%2F3d93fd8f-780a-4528-9a75-92b66591440b%2F0wvya8q_processed.png&w=3840&q=75)
Transcribed Image Text:In Problems 1-20, determine the Laplace transform of the
given function using Table 7.1 on page 356 and the properties
of the transform given in Table 7.2. [Hint: In Problems 12-20,
use an appropriate trigonometric identity.]
1. t² + e¹ sin 2t
3. e¹cos 3t+e6t1
5. 2t²e¹t+ cos 4t
7. (t-1)4
9. et sin 2t
11. cosh bt
13. sin² t
15. cos³ t
17. sin 2t sin 5t
19. cos nt sin mt, m # n
2. 31²-21
4. 3t2t² + 1
6. e 2¹ sin2t + e³4²
8. (1+e¯¹)²
10. te²¹ cos 5t
12. sin 3t cos 3t
t
14. etsin²
16. t sin² t
18. cosnt cos mt, mn
20. t sin 2t sin 5t
21. Given that L{cos bt} (s) = s/(s² + b²), use the trans-
lation property to compute L{e" cos bt}.
22. Starting with the transform L{1}(s) 1/s, use for-
mula (6) for the derivatives of the Laplace transform
to show that L{t}(s) = 1/s², L{t²} (s) = 2!/s³,
and, by using induction, that L{f} (s) = n!/s"+¹,
n = 1, 2, ....
=
23. Use Theorem 4 on page 362 to show how entry 32 fol-
lows from entry 31 in the Laplace transform table on the
inside back cover of the text.
24. Show that Leªth} (s) = n!/(s-a)"+¹ in two ways:
(a) Use the translation property for F(s).
(b) Use formula (6) for the derivatives of the Laplace
transform.
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