Use the Laplace transform to solve the difference equation 3x(t) — 4x(t − 1) = 1, x(t)=0 for t < 0.
Use the Laplace transform to solve the difference equation 3x(t) — 4x(t − 1) = 1, x(t)=0 for t < 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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![**Solving Difference Equations Using the Laplace Transform**
In this section, we will explore how the Laplace transform can be employed to solve difference equations, which are a common occurrence in various fields such as control systems and signal processing.
**Problem Statement:**
We need to solve the following difference equation using the Laplace transform:
\[
3x(t) - 4x(t-1) = 1, \quad x(t) = 0 \quad \text{for } t < 0.
\]
**Understanding the Equation:**
1. **Given Equation:**
- \(3x(t) - 4x(t-1) = 1\)
- This is a typical first-order linear difference equation.
2. **Initial Condition:**
- \(x(t) = 0\) for \(t < 0\)
- Indicates that the system is initially at rest.
**Steps to Solve Using Laplace Transform:**
1. **Apply the Laplace Transform:**
- Convert the given equation into the Laplace domain. Utilize properties of the Laplace transform for time-shifted functions.
2. **Solve Algebraically:**
- Rearrange and solve the algebraic equation in the Laplace domain.
3. **Inverse Laplace Transform:**
- Convert the solution back into the time domain to find \(x(t)\).
We will demonstrate each step with detailed computations and explanations further down this page. Solving such equations allows us to understand system behaviors over time, particularly when initial conditions and external inputs are known.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd104ea5e-59f4-47a3-a0d3-8c9f1ad3b57c%2F1d507dea-94eb-4bb2-80ee-4cf856a6acf6%2F780d17h_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving Difference Equations Using the Laplace Transform**
In this section, we will explore how the Laplace transform can be employed to solve difference equations, which are a common occurrence in various fields such as control systems and signal processing.
**Problem Statement:**
We need to solve the following difference equation using the Laplace transform:
\[
3x(t) - 4x(t-1) = 1, \quad x(t) = 0 \quad \text{for } t < 0.
\]
**Understanding the Equation:**
1. **Given Equation:**
- \(3x(t) - 4x(t-1) = 1\)
- This is a typical first-order linear difference equation.
2. **Initial Condition:**
- \(x(t) = 0\) for \(t < 0\)
- Indicates that the system is initially at rest.
**Steps to Solve Using Laplace Transform:**
1. **Apply the Laplace Transform:**
- Convert the given equation into the Laplace domain. Utilize properties of the Laplace transform for time-shifted functions.
2. **Solve Algebraically:**
- Rearrange and solve the algebraic equation in the Laplace domain.
3. **Inverse Laplace Transform:**
- Convert the solution back into the time domain to find \(x(t)\).
We will demonstrate each step with detailed computations and explanations further down this page. Solving such equations allows us to understand system behaviors over time, particularly when initial conditions and external inputs are known.
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