a) f(t) = 2tOREM (0) c) f(t) = sint e) f(t) =< 4, 0≤t≤2 3, 2= {1 t, 1
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
1a
![### Functions
This section covers a variety of mathematical functions, each presented with their respective equations:
**a) Linear Function:**
\[ f(t) = 2t \]
This function represents a linear relationship where the output is twice the input value.
**b) Exponential Function:**
\[ f(t) = e^{-3t} \]
This function describes an exponential decay based on the exponential constant \( e \) and the input \( t \).
**c) Sine Function:**
\[ f(t) = \sin t \]
This is a periodic function representing the sine wave, which oscillates between -1 and 1.
**d) Cosine Function:**
\[ f(t) = \cos t \]
Similar to the sine function, this periodic function represents the cosine wave.
**e) Piecewise Function:**
\[ f(t) = \begin{cases}
4, & 0 \leq t \leq 2 \\
3, & 2 < t < \infty
\end{cases} \]
This function has two parts:
- A constant value of 4 for \( t \) between 0 and 2 (inclusive).
- A constant value of 3 for \( t \) greater than 2.
**f) Piecewise Function:**
\[ f(t) = \begin{cases}
1, & 0 \leq t \leq 1 \\
t, & 1 < t < \infty
\end{cases} \]
This function has two parts:
- A constant value of 1 for \( t \) between 0 and 1 (inclusive).
- An identity function for \( t \) greater than 1, where the output equals the input.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb33255d3-bc66-4cf8-b4b6-36dba3c0ca10%2Fbb3c0c5e-fa7b-4a26-b3b6-1e1e5ced8b9b%2F7wont0f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Functions
This section covers a variety of mathematical functions, each presented with their respective equations:
**a) Linear Function:**
\[ f(t) = 2t \]
This function represents a linear relationship where the output is twice the input value.
**b) Exponential Function:**
\[ f(t) = e^{-3t} \]
This function describes an exponential decay based on the exponential constant \( e \) and the input \( t \).
**c) Sine Function:**
\[ f(t) = \sin t \]
This is a periodic function representing the sine wave, which oscillates between -1 and 1.
**d) Cosine Function:**
\[ f(t) = \cos t \]
Similar to the sine function, this periodic function represents the cosine wave.
**e) Piecewise Function:**
\[ f(t) = \begin{cases}
4, & 0 \leq t \leq 2 \\
3, & 2 < t < \infty
\end{cases} \]
This function has two parts:
- A constant value of 4 for \( t \) between 0 and 2 (inclusive).
- A constant value of 3 for \( t \) greater than 2.
**f) Piecewise Function:**
\[ f(t) = \begin{cases}
1, & 0 \leq t \leq 1 \\
t, & 1 < t < \infty
\end{cases} \]
This function has two parts:
- A constant value of 1 for \( t \) between 0 and 1 (inclusive).
- An identity function for \( t \) greater than 1, where the output equals the input.
![**Problem 1:**
Determine the Laplace transform of the given function by solving the improper integral; i.e., using the definition of the Laplace transform.
**Explanation:**
The Laplace transform is an integral transform that converts a function of a real variable \( t \) (often time) to a function of a complex variable \( s \) (complex frequency). It is an important tool in engineering and physics for solving differential equations.
The Laplace transform \( \mathcal{L}\{f(t)\} \) is defined by the integral:
\[ \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt \]
where:
- \( f(t) \) is the original function,
- \( s \) is a complex number,
- \( e \) is the base of the natural logarithm.
In this problem, you are asked to compute the Laplace transform using this definition, which involves evaluating an improper integral from 0 to infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb33255d3-bc66-4cf8-b4b6-36dba3c0ca10%2Fbb3c0c5e-fa7b-4a26-b3b6-1e1e5ced8b9b%2Feg7fzy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1:**
Determine the Laplace transform of the given function by solving the improper integral; i.e., using the definition of the Laplace transform.
**Explanation:**
The Laplace transform is an integral transform that converts a function of a real variable \( t \) (often time) to a function of a complex variable \( s \) (complex frequency). It is an important tool in engineering and physics for solving differential equations.
The Laplace transform \( \mathcal{L}\{f(t)\} \) is defined by the integral:
\[ \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt \]
where:
- \( f(t) \) is the original function,
- \( s \) is a complex number,
- \( e \) is the base of the natural logarithm.
In this problem, you are asked to compute the Laplace transform using this definition, which involves evaluating an improper integral from 0 to infinity.
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