Use the Laplace transform to solve the given initial-value problem. 0, 0≤t< π y" + y = f(t), y(0) = 0, y'(0) = 1, where f(t) = 1, π ≤ t < 2π 0, t≥ 2π y(t) = + |)(x − x) + ( t- )u(t –
Use the Laplace transform to solve the given initial-value problem. 0, 0≤t< π y" + y = f(t), y(0) = 0, y'(0) = 1, where f(t) = 1, π ≤ t < 2π 0, t≥ 2π y(t) = + |)(x − x) + ( t- )u(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
![**Problem Statement:**
Use the Laplace transform to solve the given initial-value problem.
\[ y'' + y = f(t), \quad y(0) = 0, \, y'(0) = 1, \]
where
\[
f(t) =
\begin{cases}
0, & 0 \leq t < \pi \\
1, & \pi \leq t < 2\pi \\
0, & t \geq 2\pi
\end{cases}
\]
**Solution Representation:**
\[
y(t) = \boxed{\phantom{x}} + \left( \boxed{\phantom{x}} \right) u(t - \pi) + \left( \boxed{\phantom{x}} \right) u(t - \boxed{\phantom{x}})
\]
**Explanation of Content:**
The problem focuses on using the Laplace transform to solve a second-order linear differential equation with piecewise-defined function \( f(t) \). The given function \( f(t) \) acts as a switching function based on intervals of \( t \). The solution is expressed in terms of step functions, denoted by \( u(t - c) \), which represent the unit step function shifted by \( c \).
The challenge involves finding the correct terms, placing them in the boxed areas, and understanding their contributions in different intervals determined by the step functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccae5729-7489-41ed-a32a-24cdd49b7598%2Fd2ee61d1-31db-4f0f-b305-ece690101023%2F1zu5irs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the Laplace transform to solve the given initial-value problem.
\[ y'' + y = f(t), \quad y(0) = 0, \, y'(0) = 1, \]
where
\[
f(t) =
\begin{cases}
0, & 0 \leq t < \pi \\
1, & \pi \leq t < 2\pi \\
0, & t \geq 2\pi
\end{cases}
\]
**Solution Representation:**
\[
y(t) = \boxed{\phantom{x}} + \left( \boxed{\phantom{x}} \right) u(t - \pi) + \left( \boxed{\phantom{x}} \right) u(t - \boxed{\phantom{x}})
\]
**Explanation of Content:**
The problem focuses on using the Laplace transform to solve a second-order linear differential equation with piecewise-defined function \( f(t) \). The given function \( f(t) \) acts as a switching function based on intervals of \( t \). The solution is expressed in terms of step functions, denoted by \( u(t - c) \), which represent the unit step function shifted by \( c \).
The challenge involves finding the correct terms, placing them in the boxed areas, and understanding their contributions in different intervals determined by the step functions.
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