-¹ {5 (a) 2-1 [ 3)²} (b) L-1 s(s+3)² { (3² + 4)²} (Hint: you may want to use the trig identity sin(a) cos(b) = sin(a+b)+sin(a−b))
-¹ {5 (a) 2-1 [ 3)²} (b) L-1 s(s+3)² { (3² + 4)²} (Hint: you may want to use the trig identity sin(a) cos(b) = sin(a+b)+sin(a−b))
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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![**Inverse Laplace Transform as Convolution Integrals**
**Objective:**
Evaluate the given inverse Laplace transforms as convolution integrals, and then evaluate the integrals without directly referencing the table of Laplace transforms.
---
**Exercises:**
(a) Evaluate the inverse Laplace transform:
\[
\mathcal{L}^{-1} \left\{ \frac{1}{s(s + 3)^2} \right\}
\]
(b) Evaluate the inverse Laplace transform:
\[
\mathcal{L}^{-1} \left\{ \frac{s}{(s^2 + 4)^2} \right\}
\]
**Hint:** For part (b), you may want to use the trigonometric identity:
\[
\sin(a) \cos(b) = \frac{1}{2} \sin(a+b) + \frac{1}{2} \sin(a-b)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74a50780-bdf2-45e2-b018-f4cc84bd693f%2Fce4d174b-bfc3-4a23-873d-9af5948d5f9b%2Febg022w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Inverse Laplace Transform as Convolution Integrals**
**Objective:**
Evaluate the given inverse Laplace transforms as convolution integrals, and then evaluate the integrals without directly referencing the table of Laplace transforms.
---
**Exercises:**
(a) Evaluate the inverse Laplace transform:
\[
\mathcal{L}^{-1} \left\{ \frac{1}{s(s + 3)^2} \right\}
\]
(b) Evaluate the inverse Laplace transform:
\[
\mathcal{L}^{-1} \left\{ \frac{s}{(s^2 + 4)^2} \right\}
\]
**Hint:** For part (b), you may want to use the trigonometric identity:
\[
\sin(a) \cos(b) = \frac{1}{2} \sin(a+b) + \frac{1}{2} \sin(a-b)
\]
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