Consider the initial value problem y" + (1/3)y' + 4y = fk(t), y(0) = y'(0) = 0,
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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![To solve the initial value problem, consider the following differential equation:
\[ y'' + \left(\frac{1}{3}\right)y' + 4y = f_k(t) \]
where the initial conditions are given by:
\[ y(0) = 0, \quad y'(0) = 0 \]
This problem involves finding the function \( y(t) \) that satisfies the differential equation with the specified initial conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ea01508-f9b6-4a19-b038-56e12c298daf%2Fd296165d-83af-41fb-9217-ca7bc300d7b0%2Fhfn70f4_processed.png&w=3840&q=75)
Transcribed Image Text:To solve the initial value problem, consider the following differential equation:
\[ y'' + \left(\frac{1}{3}\right)y' + 4y = f_k(t) \]
where the initial conditions are given by:
\[ y(0) = 0, \quad y'(0) = 0 \]
This problem involves finding the function \( y(t) \) that satisfies the differential equation with the specified initial conditions.
![### Problem Statement
Consider the function \( f_k(t) \) defined as:
\[
f_k(t) =
\begin{cases}
1/2k, & 4 - k \leq t \leq 4 + k \\
0, & 0 \leq t < 4 - k, \ t \geq 4 + k
\end{cases}
\]
### Tasks
#### (a) Graph Sketching
- **Objective:** Sketch the graph of \( f_k(t) \).
- **Analysis:** Show that the area under the graph of \( f_k(t) \) is independent of \( k \).
#### (b) Heaviside Functions
- **Objective:** Express \( f_k(t) \) using Heaviside functions.
- **Further Task:** Solve the associated differential equation.
#### (c) Computer Plotting
- **Objective:** Using a computer, plot the solutions for \( k = 2 \) and \( k = \frac{1}{2} \).
- **Analysis:** Describe how the solution depends on \( k \).
### Explanation of Graphical Representation
The graph of \( f_k(t) \) consists of a horizontal line segment at the height \( \frac{1}{2k} \) over the interval \([4-k, 4+k]\). Outside this interval, the function is zero.
- **Area Independence:** The calculation involved in illustrating the area independence involves integrating over the interval where \( f_k(t) \) is non-zero. Given the piecewise definition, adjusting \( k \) rescales the width of the interval but inversely alters the height of the line segment, keeping the overall area constant.
**Note:** When plotting the solutions, adjust \( k \) to visualize its impact on the width of the function's non-zero interval and the function's height.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ea01508-f9b6-4a19-b038-56e12c298daf%2Fd296165d-83af-41fb-9217-ca7bc300d7b0%2Fitwgi9_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Consider the function \( f_k(t) \) defined as:
\[
f_k(t) =
\begin{cases}
1/2k, & 4 - k \leq t \leq 4 + k \\
0, & 0 \leq t < 4 - k, \ t \geq 4 + k
\end{cases}
\]
### Tasks
#### (a) Graph Sketching
- **Objective:** Sketch the graph of \( f_k(t) \).
- **Analysis:** Show that the area under the graph of \( f_k(t) \) is independent of \( k \).
#### (b) Heaviside Functions
- **Objective:** Express \( f_k(t) \) using Heaviside functions.
- **Further Task:** Solve the associated differential equation.
#### (c) Computer Plotting
- **Objective:** Using a computer, plot the solutions for \( k = 2 \) and \( k = \frac{1}{2} \).
- **Analysis:** Describe how the solution depends on \( k \).
### Explanation of Graphical Representation
The graph of \( f_k(t) \) consists of a horizontal line segment at the height \( \frac{1}{2k} \) over the interval \([4-k, 4+k]\). Outside this interval, the function is zero.
- **Area Independence:** The calculation involved in illustrating the area independence involves integrating over the interval where \( f_k(t) \) is non-zero. Given the piecewise definition, adjusting \( k \) rescales the width of the interval but inversely alters the height of the line segment, keeping the overall area constant.
**Note:** When plotting the solutions, adjust \( k \) to visualize its impact on the width of the function's non-zero interval and the function's height.
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