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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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![### Problem Statement
**a) Write the non-homogeneous term \( f(t) \) in terms of unit step functions:**
\[
f(t) =
\begin{cases}
0 & \text{if } t < 1, \\
1 & \text{if } 1 \leq t < 2, \\
-1 & \text{if } t \geq 2
\end{cases}
\]
**b) Use your answer to (a) to find the Laplace Transform of \( f(t) \) using linearity and the table of Laplace Transforms.**
### Explanation
- **Unit Step Function Representation:** The unit step function, often denoted \( u(t-a) \), equals zero for \( t < a \) and one for \( t \geq a \). It’s used to "turn on" or "turn off" parts of a piecewise function.
- **Laplace Transform:** A technique employed in engineering and physics to transform complex differential equations into simpler algebraic equations, defined as:
\[
\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt
\]
where \( s \) is a complex number. This transform helps in solving differential equations.
In this case, you need to express \( f(t) \) with unit step functions and then apply the Laplace Transform to solve it.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa9a9ea7-0a60-40f2-88c8-00f846edd995%2Fc8a47986-3532-45b3-8875-6b25e9f33f4f%2Fd7xfis_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**a) Write the non-homogeneous term \( f(t) \) in terms of unit step functions:**
\[
f(t) =
\begin{cases}
0 & \text{if } t < 1, \\
1 & \text{if } 1 \leq t < 2, \\
-1 & \text{if } t \geq 2
\end{cases}
\]
**b) Use your answer to (a) to find the Laplace Transform of \( f(t) \) using linearity and the table of Laplace Transforms.**
### Explanation
- **Unit Step Function Representation:** The unit step function, often denoted \( u(t-a) \), equals zero for \( t < a \) and one for \( t \geq a \). It’s used to "turn on" or "turn off" parts of a piecewise function.
- **Laplace Transform:** A technique employed in engineering and physics to transform complex differential equations into simpler algebraic equations, defined as:
\[
\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt
\]
where \( s \) is a complex number. This transform helps in solving differential equations.
In this case, you need to express \( f(t) \) with unit step functions and then apply the Laplace Transform to solve it.
![The differential equation to be solved is:
\[ y'' - 3y' + 2y =
\begin{cases}
0, & \text{if } 0 \leq t < 1, \\
1, & \text{if } 1 \leq t < 2, \\
-1, & \text{if } t \geq 2
\end{cases} \]
with the initial conditions:
\[ y(0) = 0, \]
\[ y'(0) = 0. \]
This piecewise function defines three different forcing terms over specific intervals of \( t \). Each interval suggests a different constant term in the differential equation, affecting the solution within that interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa9a9ea7-0a60-40f2-88c8-00f846edd995%2Fc8a47986-3532-45b3-8875-6b25e9f33f4f%2F8vc5p28_processed.png&w=3840&q=75)
Transcribed Image Text:The differential equation to be solved is:
\[ y'' - 3y' + 2y =
\begin{cases}
0, & \text{if } 0 \leq t < 1, \\
1, & \text{if } 1 \leq t < 2, \\
-1, & \text{if } t \geq 2
\end{cases} \]
with the initial conditions:
\[ y(0) = 0, \]
\[ y'(0) = 0. \]
This piecewise function defines three different forcing terms over specific intervals of \( t \). Each interval suggests a different constant term in the differential equation, affecting the solution within that interval.
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