Consider the following initial value problem: y" + 25y = f(t); f(t) = { y(0) = 0, y'(0) = 5; 5, 0≤t < 3³ 3π 0, 35 ≤ t <∞ (a) Sketch the graph of the forcing function on an appropriate interval. 10+ 9 8 7 6 5 4 3 2 1 3 4 6 (b) Find the solution of the given initial value problem. NOTE: Denote the Heaviside function by ue(t) where uc(t) = 1 if t≥c and 0 otherwise. Indicate separately the value of c. y(t) =
Consider the following initial value problem: y" + 25y = f(t); f(t) = { y(0) = 0, y'(0) = 5; 5, 0≤t < 3³ 3π 0, 35 ≤ t <∞ (a) Sketch the graph of the forcing function on an appropriate interval. 10+ 9 8 7 6 5 4 3 2 1 3 4 6 (b) Find the solution of the given initial value problem. NOTE: Denote the Heaviside function by ue(t) where uc(t) = 1 if t≥c and 0 otherwise. Indicate separately the value of c. y(t) =
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
![Consider the following initial value problem:
\[ y'' + 25y = f(t); \quad y(0) = 0, \quad y'(0) = 5; \]
\[
f(t) =
\begin{cases}
5, & 0 \leq t < \frac{3\pi}{5} \\
0, & \frac{3\pi}{5} \leq t < \infty
\end{cases}
\]
(a) Sketch the graph of the forcing function on an appropriate interval.
**Graph Description:**
- The graph is a piecewise function illustrating \( f(t) \).
- From \( t = 0 \) to \( t = \frac{3\pi}{5} \), the value of \( f(t) \) is 5. This is shown as a horizontal line at 5 between these two points.
- At \( t = \frac{3\pi}{5} \), there is an open circle indicating the function does not include this point.
- For \( t \geq \frac{3\pi}{5} \), \( f(t) \) is 0, shown as a horizontal line along the t-axis.
- The graph visually emphasizes the discontinuity and transition at \( t = \frac{3\pi}{5} \).
(b) Find the solution of the given initial value problem.
**NOTE:** Denote the Heaviside function by \( u_c(t) \) where \( u_c(t) = 1 \) if \( t \geq c \) and 0 otherwise. Indicate separately the value of \( c \).
\[ y(t) = \]
*[The solution would typically involve solving the differential equation with the specified initial conditions and using the Heaviside function where appropriate.]*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c836437-a2a4-49ad-8637-6abf33735fd4%2Fc1d82a45-dd56-48db-bba5-b625fb619a4a%2F43xy7d5_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following initial value problem:
\[ y'' + 25y = f(t); \quad y(0) = 0, \quad y'(0) = 5; \]
\[
f(t) =
\begin{cases}
5, & 0 \leq t < \frac{3\pi}{5} \\
0, & \frac{3\pi}{5} \leq t < \infty
\end{cases}
\]
(a) Sketch the graph of the forcing function on an appropriate interval.
**Graph Description:**
- The graph is a piecewise function illustrating \( f(t) \).
- From \( t = 0 \) to \( t = \frac{3\pi}{5} \), the value of \( f(t) \) is 5. This is shown as a horizontal line at 5 between these two points.
- At \( t = \frac{3\pi}{5} \), there is an open circle indicating the function does not include this point.
- For \( t \geq \frac{3\pi}{5} \), \( f(t) \) is 0, shown as a horizontal line along the t-axis.
- The graph visually emphasizes the discontinuity and transition at \( t = \frac{3\pi}{5} \).
(b) Find the solution of the given initial value problem.
**NOTE:** Denote the Heaviside function by \( u_c(t) \) where \( u_c(t) = 1 \) if \( t \geq c \) and 0 otherwise. Indicate separately the value of \( c \).
\[ y(t) = \]
*[The solution would typically involve solving the differential equation with the specified initial conditions and using the Heaviside function where appropriate.]*
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