Use the Laplace transform to solve the initial value problem Note: Use u for the step function. y'' + 2y' + y = et + 28(t− 2), y(0) = − 1, y'(0) = 2
Use the Laplace transform to solve the initial value problem Note: Use u for the step function. y'' + 2y' + y = et + 28(t− 2), y(0) = − 1, y'(0) = 2
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 Initial Value Problems Using Laplace Transforms
In this section, we will learn how to use the Laplace transform to solve a given initial value problem.
Consider the following differential equation and initial conditions:
\[ y'' + 2y' + y = e^t + 2\delta(t - 2), \]
\[ y(0) = -1, \quad y'(0) = 2. \]
**Note:** Use \(u\) for the step function.
Here, we are given a second-order linear differential equation with a non-homogeneous term that includes an exponential function \(e^t\) and a delta function \(2\delta(t - 2)\). The initial conditions provided are \(y(0) = -1\) and \(y'(0) = 2\).
### Steps to Solve the Problem
1. **Apply the Laplace Transform:**
- Take the Laplace transform of both sides of the differential equation.
- Utilize the properties of the Laplace transform for the given functions and initial conditions.
2. **Solve for \(Y(s)\):**
- Express the transformed equation in terms of the Laplace variable \(s\).
- Solve for \(Y(s)\), the Laplace transform of \(y(t)\).
3. **Inverse Laplace Transform:**
- Use the inverse Laplace transform to convert \(Y(s)\) back to \(y(t)\).
These steps will guide you through solving the initial value problem by breaking it down into manageable parts. This method leverages the powerful properties of the Laplace transform to handle the initial conditions and complex non-homogeneous terms efficiently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3733a5-5e9e-433b-b6d1-2cfec636672d%2F6bfa59be-2117-4d54-9510-7a281404afcd%2F11soretc_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Initial Value Problems Using Laplace Transforms
In this section, we will learn how to use the Laplace transform to solve a given initial value problem.
Consider the following differential equation and initial conditions:
\[ y'' + 2y' + y = e^t + 2\delta(t - 2), \]
\[ y(0) = -1, \quad y'(0) = 2. \]
**Note:** Use \(u\) for the step function.
Here, we are given a second-order linear differential equation with a non-homogeneous term that includes an exponential function \(e^t\) and a delta function \(2\delta(t - 2)\). The initial conditions provided are \(y(0) = -1\) and \(y'(0) = 2\).
### Steps to Solve the Problem
1. **Apply the Laplace Transform:**
- Take the Laplace transform of both sides of the differential equation.
- Utilize the properties of the Laplace transform for the given functions and initial conditions.
2. **Solve for \(Y(s)\):**
- Express the transformed equation in terms of the Laplace variable \(s\).
- Solve for \(Y(s)\), the Laplace transform of \(y(t)\).
3. **Inverse Laplace Transform:**
- Use the inverse Laplace transform to convert \(Y(s)\) back to \(y(t)\).
These steps will guide you through solving the initial value problem by breaking it down into manageable parts. This method leverages the powerful properties of the Laplace transform to handle the initial conditions and complex non-homogeneous terms efficiently.
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