Solve the third-order initial value problem below using the method of Laplace transforms. y'"' + 7y" + 7y' - 15y = -75, y(0) = 8, y'(0) = -9, y''(0) = 75 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 third-order initial value problem below using the method of Laplace transforms. y'"' + 7y" + 7y' - 15y = -75, y(0) = 8, y'(0) = -9, y''(0) = 75 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 Third-Order Initial Value Problems Using Laplace Transforms**
On this page, we will explore the process of solving a third-order initial value problem using Laplace transforms. Here is the problem statement:
\[ y''' + 7y'' + 7y' - 15y = -75, \]
with the initial conditions:
\[ y(0) = 8, \quad y'(0) = -9, \quad y''(0) = 75. \]
To assist you in solving this type of problem, we provide two essential resources:
1. [Table of Laplace Transforms](#)
2. [Table of Properties of Laplace Transforms](#)
Below, you can type your solution for \( y(t) \):
\[ y(t) = \boxed{\hspace{5cm}} \]
(Type an exact answer in terms of \( e \).)
**How to approach this problem:**
1. Start by taking the Laplace transform of each term in the differential equation.
2. Use the initial conditions to substitute into the transformed equation.
3. Solve the resulting algebraic equation for \( Y(s) \).
4. Apply the inverse Laplace transform to find \( y(t) \).
By following these steps, you can systematically solve third-order initial value problems involving differential equations. Remember to make use of the tables provided to assist with the transformations.
For further details and examples, click the links above to view the tables of Laplace transforms and their properties. These resources provide comprehensive information on how different functions and their derivatives transform under the Laplace operation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45088ba1-a1a3-4a96-b462-54f2c0a5a29e%2F7f3cc8e4-a497-41b7-ab4d-a37765372323%2Fyenl3lc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving Third-Order Initial Value Problems Using Laplace Transforms**
On this page, we will explore the process of solving a third-order initial value problem using Laplace transforms. Here is the problem statement:
\[ y''' + 7y'' + 7y' - 15y = -75, \]
with the initial conditions:
\[ y(0) = 8, \quad y'(0) = -9, \quad y''(0) = 75. \]
To assist you in solving this type of problem, we provide two essential resources:
1. [Table of Laplace Transforms](#)
2. [Table of Properties of Laplace Transforms](#)
Below, you can type your solution for \( y(t) \):
\[ y(t) = \boxed{\hspace{5cm}} \]
(Type an exact answer in terms of \( e \).)
**How to approach this problem:**
1. Start by taking the Laplace transform of each term in the differential equation.
2. Use the initial conditions to substitute into the transformed equation.
3. Solve the resulting algebraic equation for \( Y(s) \).
4. Apply the inverse Laplace transform to find \( y(t) \).
By following these steps, you can systematically solve third-order initial value problems involving differential equations. Remember to make use of the tables provided to assist with the transformations.
For further details and examples, click the links above to view the tables of Laplace transforms and their properties. These resources provide comprehensive information on how different functions and their derivatives transform under the Laplace operation.
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