Solve the initial value problem below using the method of Laplace transforms. y" - 4y' + 40y = 185 e¹, y(0) = 5, y'(0) = 11 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. y(t) = (Type an exact answer in terms of e.)
Solve the initial value problem below using the method of Laplace transforms. y" - 4y' + 40y = 185 e¹, y(0) = 5, y'(0) = 11 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. y(t) = (Type an exact answer in terms of e.)
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**
**Example Problem:**
Solve the initial value problem below using the method of Laplace transforms.
\[ y'' - 4y' + 4y = 185 e^t \]
\[ y(0) = 5, \quad y'(0) = 11 \]
**Resources:**
- [Click here to view the table of Laplace transforms.](#)
- [Click here to view the table of properties of Laplace transforms.](#)
**Solution:**
\[ y(t) = \_\_\_\_ \]
*(Type an exact answer in terms of \( e \).)*
---
### Explanation:
To solve this differential equation using Laplace transforms:
1. **Take the Laplace transform** of both sides of the differential equation.
2. **Apply the initial conditions** \( y(0) = 5 \) and \( y'(0) = 11 \) to solve for the unknowns.
3. **Find the inverse Laplace transform** of the resulting expression to determine \( y(t) \).
This method can be very efficient for solving linear ordinary differential equations with given initial conditions, especially when forcing functions like \( e^t \) are involved. Use the provided tables of Laplace transforms and their properties to assist in these steps.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45088ba1-a1a3-4a96-b462-54f2c0a5a29e%2F9b891653-cade-4fc1-b33a-e3abf604c497%2Fbrmqz1f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving Initial Value Problems Using Laplace Transforms**
**Example Problem:**
Solve the initial value problem below using the method of Laplace transforms.
\[ y'' - 4y' + 4y = 185 e^t \]
\[ y(0) = 5, \quad y'(0) = 11 \]
**Resources:**
- [Click here to view the table of Laplace transforms.](#)
- [Click here to view the table of properties of Laplace transforms.](#)
**Solution:**
\[ y(t) = \_\_\_\_ \]
*(Type an exact answer in terms of \( e \).)*
---
### Explanation:
To solve this differential equation using Laplace transforms:
1. **Take the Laplace transform** of both sides of the differential equation.
2. **Apply the initial conditions** \( y(0) = 5 \) and \( y'(0) = 11 \) to solve for the unknowns.
3. **Find the inverse Laplace transform** of the resulting expression to determine \( y(t) \).
This method can be very efficient for solving linear ordinary differential equations with given initial conditions, especially when forcing functions like \( e^t \) are involved. Use the provided tables of Laplace transforms and their properties to assist in these steps.
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