Use the Laplace transform to solve the initial-value problem, y" + 4y = 1-u(t − 1); y(0) = 0, y'(0) = -1
Use the Laplace transform to solve the initial-value problem, y" + 4y = 1-u(t − 1); y(0) = 0, y'(0) = -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
Use the Laplace transform to solve the initial-value problem,
?′′ + 4? = 1 − ?(? − 1); ?(0) = 0, ?′(0) = −1
![**Problem 7.3: Laplace Transform for Initial-Value Problems**
Use the Laplace transform to solve the initial-value problem:
\[ y'' + 4y = 1 - u(t - 1) \]
with the initial conditions:
\[ y(0) = 0, \quad y'(0) = -1 \]
**Explanation**:
- **\( y'' + 4y = 1 - u(t - 1) \)**: This is a linear second-order differential equation with a discontinuous forcing function described by the unit step function \( u(t - 1) \).
- **\( y(0) = 0 \), \( y'(0) = -1 \)**: These are the initial conditions needed to solve the differential equation using Laplace transforms. Here, \(y(0)\) is the initial position, and \(y'(0)\) is the initial velocity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdaa88007-4c38-49b3-b187-d2dfdab0e7df%2F6d3b686c-5c9d-4b33-b013-f24c34527406%2Fhp1brg4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 7.3: Laplace Transform for Initial-Value Problems**
Use the Laplace transform to solve the initial-value problem:
\[ y'' + 4y = 1 - u(t - 1) \]
with the initial conditions:
\[ y(0) = 0, \quad y'(0) = -1 \]
**Explanation**:
- **\( y'' + 4y = 1 - u(t - 1) \)**: This is a linear second-order differential equation with a discontinuous forcing function described by the unit step function \( u(t - 1) \).
- **\( y(0) = 0 \), \( y'(0) = -1 \)**: These are the initial conditions needed to solve the differential equation using Laplace transforms. Here, \(y(0)\) is the initial position, and \(y'(0)\) is the initial velocity.
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