3. Find the Laplace transform of r(t) = sin kt and x(t): definitions sin kt = 1 ( (eikt - e-ikt), coskt = 2 = cos kt using the 2i Check your answers in the table. (eikt + e-ikt).
3. Find the Laplace transform of r(t) = sin kt and x(t): definitions sin kt = 1 ( (eikt - e-ikt), coskt = 2 = cos kt using the 2i Check your answers in the table. (eikt + e-ikt).
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
Please do 3
![# 3.1 Definition and Basic Properties
### Exercises
2. Use the definition of the Laplace transform to find the transform of \( x(t) = e^{-3t} H(t - 2) \).
3. Find the Laplace transform of \( x(t) = \sin kt \) and \( x(t) = \cos kt \) using the definitions
\[
\sin kt = \frac{1}{2i} (e^{ikt} - e^{-ikt}), \quad \cos kt = \frac{1}{2}(e^{ikt} + e^{-ikt}).
\]
Check your answers in the table.
4. Find the Laplace transform of the hyperbolic functions \( x(t) = \sinh kt \) and \( x(t) = \cosh kt \) using the definitions
\[
\sinh kt = \frac{1}{2}(e^{kt} - e^{-kt}), \quad \cosh kt = \frac{1}{2}(e^{kt} + e^{-kt}).
\]
Check your answers in the table.
5. Derive the operational formulas (3.4) and (3.5) directly from the definition.
**Hint:** Change variables in the integrals.
6. Use the definition of the Laplace transform to show that
\[
\mathcal{L}[f(t)H(t - a)] = e^{-as}\mathcal{L}[f(t + a)].
\]
7. Use the preceding exercise to compute \( \mathcal{L}[t^2 H(t - 1)] \).
8. Find the Laplace transform of the following functions.
a) \( 6 + 5e^{-2t} + te^{3t} \).
b) \( tH(t - 3) \).
c) \( \cos 5t \).
d) \( \sin(2t + \pi) \).
e) \( 3e^{-t} \cosh t \).
f) \( H(t - \pi) \cos(t - \pi) \).
9. Find the inverse transform of the following functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc10cce41-ef51-4545-a724-9736e236d3b7%2F368d1523-e5b0-4ed8-b016-a8a68badda74%2Fimklexp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# 3.1 Definition and Basic Properties
### Exercises
2. Use the definition of the Laplace transform to find the transform of \( x(t) = e^{-3t} H(t - 2) \).
3. Find the Laplace transform of \( x(t) = \sin kt \) and \( x(t) = \cos kt \) using the definitions
\[
\sin kt = \frac{1}{2i} (e^{ikt} - e^{-ikt}), \quad \cos kt = \frac{1}{2}(e^{ikt} + e^{-ikt}).
\]
Check your answers in the table.
4. Find the Laplace transform of the hyperbolic functions \( x(t) = \sinh kt \) and \( x(t) = \cosh kt \) using the definitions
\[
\sinh kt = \frac{1}{2}(e^{kt} - e^{-kt}), \quad \cosh kt = \frac{1}{2}(e^{kt} + e^{-kt}).
\]
Check your answers in the table.
5. Derive the operational formulas (3.4) and (3.5) directly from the definition.
**Hint:** Change variables in the integrals.
6. Use the definition of the Laplace transform to show that
\[
\mathcal{L}[f(t)H(t - a)] = e^{-as}\mathcal{L}[f(t + a)].
\]
7. Use the preceding exercise to compute \( \mathcal{L}[t^2 H(t - 1)] \).
8. Find the Laplace transform of the following functions.
a) \( 6 + 5e^{-2t} + te^{3t} \).
b) \( tH(t - 3) \).
c) \( \cos 5t \).
d) \( \sin(2t + \pi) \).
e) \( 3e^{-t} \cosh t \).
f) \( H(t - \pi) \cos(t - \pi) \).
9. Find the inverse transform of the following functions.
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