Use the Laplace transform to solve the given initial-value problem. y" +9y' = 8(t− 1), y(0) = 0, y'(0) = 1
Use the Laplace transform to solve the given initial-value problem. y" +9y' = 8(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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![### Solving Initial-Value Problems Using Laplace Transform
To solve the given initial-value problem using the Laplace transform, follow the steps and calculations below:
#### Problem Statement
Solve the differential equation:
\[ y'' + 9y' = \delta(t - 1) \]
with initial conditions:
\[ y(0) = 0, \quad y'(0) = 1 \]
#### Solution
We apply the Laplace transform technique which involves the following steps:
1. **Transform the Differential Equation**:
The Laplace transform is used to convert the differential equation into an algebraic equation.
2. **Solve the Algebraic Equation**:
Solve the resulting algebraic equation for the Laplace transform of the solution.
3. **Inverse Laplace Transform**:
Finally, calculate the inverse Laplace transform to obtain the solution in the time domain.
The solution to the initial-value problem is:
\[ y(t) = \left( \frac{1}{9} \left( 1 - e^{-9(t-1)} \right) \right) + \left( \left( \frac{1}{9} \left( 1 - e^{-9t} \right) \right) \cdot u(t - 1) \right) \]
#### Diagram Explanation
- The boxed expressions represent components of the solution function \( y(t) \).
- The \( \delta(t - 1) \) represents the forcing function applied at \( t = 1 \).
- The terms involving the exponential functions \( e^{-9(t-1)} \) and \( e^{-9t} \) describe the system's response over time.
#### Additional Information
**Need Help?**
- **Read It**: Click this button for a detailed explanation of the Laplace transform method.
- **Master It**: Click this button to practice more examples and master solving differential equations using Laplace transforms.
This approach helps students to understand and apply the Laplace transform in solving differential equations with initial conditions effectively.
For further assistance, students can access additional resources or ask questions in the help section.
---
By following these steps and utilizing the resources provided, students can gain a deeper understanding of the Laplace transform method in solving differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96023f4d-6281-4c81-803e-adfe3e4197d6%2F1fffba86-304e-421e-803a-89521c95dac7%2Ffvzvipi_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Initial-Value Problems Using Laplace Transform
To solve the given initial-value problem using the Laplace transform, follow the steps and calculations below:
#### Problem Statement
Solve the differential equation:
\[ y'' + 9y' = \delta(t - 1) \]
with initial conditions:
\[ y(0) = 0, \quad y'(0) = 1 \]
#### Solution
We apply the Laplace transform technique which involves the following steps:
1. **Transform the Differential Equation**:
The Laplace transform is used to convert the differential equation into an algebraic equation.
2. **Solve the Algebraic Equation**:
Solve the resulting algebraic equation for the Laplace transform of the solution.
3. **Inverse Laplace Transform**:
Finally, calculate the inverse Laplace transform to obtain the solution in the time domain.
The solution to the initial-value problem is:
\[ y(t) = \left( \frac{1}{9} \left( 1 - e^{-9(t-1)} \right) \right) + \left( \left( \frac{1}{9} \left( 1 - e^{-9t} \right) \right) \cdot u(t - 1) \right) \]
#### Diagram Explanation
- The boxed expressions represent components of the solution function \( y(t) \).
- The \( \delta(t - 1) \) represents the forcing function applied at \( t = 1 \).
- The terms involving the exponential functions \( e^{-9(t-1)} \) and \( e^{-9t} \) describe the system's response over time.
#### Additional Information
**Need Help?**
- **Read It**: Click this button for a detailed explanation of the Laplace transform method.
- **Master It**: Click this button to practice more examples and master solving differential equations using Laplace transforms.
This approach helps students to understand and apply the Laplace transform in solving differential equations with initial conditions effectively.
For further assistance, students can access additional resources or ask questions in the help section.
---
By following these steps and utilizing the resources provided, students can gain a deeper understanding of the Laplace transform method in solving differential equations.
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