Solve the following o.d.e by using the method of Laplace Transform. t if 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 Differential Equations Using Laplace Transforms
**Problem Statement:**
Solve the following ordinary differential equation (O.D.E) by using the method of Laplace Transform:
\[ y'' + 3y' + 2y = \delta(t - 5) + f(t); \]
with the initial conditions:
\[ y(0) = 0, \quad y'(0) = 0 \]
where the function \( f(t) \) is defined as:
\[
f(t) =
\begin{cases}
t & \text{if } 0 \leq t < 10 \\
0 & \text{if } t \geq 10
\end{cases}
\]
### Steps to solve:
1. **Apply the Laplace Transform** to both sides of the differential equation.
2. **Use the initial conditions** and properties of the Laplace Transform to simplify the transformed equation.
3. **Solve for \( Y(s) \)**, the Laplace Transform of \( y(t) \).
4. **Apply the inverse Laplace Transform** to find \( y(t) \).
### Explanation of the Diagram:
The piecewise function \( f(t) \) is represented in a combination of conditions:
- For \( 0 \leq t < 10 \), \( f(t) \) is linear, \( f(t) = t \).
- For \( t \geq 10 \), \( f(t) \) is a constant value, \( f(t) = 0 \).
This implies that the function \( f(t) \) increases linearly from 0 to 10 and becomes zero afterward.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F093e8e2d-e528-490a-b84f-32f3af733c03%2F42879aca-3e2f-4752-b224-3da44e4fc1f2%2Fdhv8cft_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Differential Equations Using Laplace Transforms
**Problem Statement:**
Solve the following ordinary differential equation (O.D.E) by using the method of Laplace Transform:
\[ y'' + 3y' + 2y = \delta(t - 5) + f(t); \]
with the initial conditions:
\[ y(0) = 0, \quad y'(0) = 0 \]
where the function \( f(t) \) is defined as:
\[
f(t) =
\begin{cases}
t & \text{if } 0 \leq t < 10 \\
0 & \text{if } t \geq 10
\end{cases}
\]
### Steps to solve:
1. **Apply the Laplace Transform** to both sides of the differential equation.
2. **Use the initial conditions** and properties of the Laplace Transform to simplify the transformed equation.
3. **Solve for \( Y(s) \)**, the Laplace Transform of \( y(t) \).
4. **Apply the inverse Laplace Transform** to find \( y(t) \).
### Explanation of the Diagram:
The piecewise function \( f(t) \) is represented in a combination of conditions:
- For \( 0 \leq t < 10 \), \( f(t) \) is linear, \( f(t) = t \).
- For \( t \geq 10 \), \( f(t) \) is a constant value, \( f(t) = 0 \).
This implies that the function \( f(t) \) increases linearly from 0 to 10 and becomes zero afterward.
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