Find explicit functions f(t), g(t) St. L{f(t]g(t)} (s) + L {f(+1} {s} £{q(+)} (S), 1.e., show the Laplace trans form hot multiplicative. IS
Find explicit functions f(t), g(t) St. L{f(t]g(t)} (s) + L {f(+1} {s} £{q(+)} (S), 1.e., show the Laplace trans form hot multiplicative. IS
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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Question
Use Leprace Transforms
![**Laplace Transform and Multiplicativity**
**Problem Statement:**
Find explicit functions \( f(t) \) and \( g(t) \) such that:
\[
\mathcal{L}\{f(t)g(t)\}(s) \neq \mathcal{L}\{f(t)\}(s) \mathcal{L}\{g(t)\}(s)
\]
i.e., show that the Laplace transform is not multiplicative.
**Explanation:**
The problem asks us to demonstrate that the Laplace transform of the product of two functions is not equal to the product of their individual Laplace transforms. To show this, one needs to identify specific functions \( f(t) \) and \( g(t) \), where their combined Laplace transform does not equal the product of their separate Laplace transforms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b5d3f26-cda5-43e5-8223-bfa02258241c%2Ff74e1b5e-0f4b-4c8c-acaa-0f2558aa788f%2Fs6spksj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Laplace Transform and Multiplicativity**
**Problem Statement:**
Find explicit functions \( f(t) \) and \( g(t) \) such that:
\[
\mathcal{L}\{f(t)g(t)\}(s) \neq \mathcal{L}\{f(t)\}(s) \mathcal{L}\{g(t)\}(s)
\]
i.e., show that the Laplace transform is not multiplicative.
**Explanation:**
The problem asks us to demonstrate that the Laplace transform of the product of two functions is not equal to the product of their individual Laplace transforms. To show this, one needs to identify specific functions \( f(t) \) and \( g(t) \), where their combined Laplace transform does not equal the product of their separate Laplace transforms.
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