Solve for Y(s), the Laplace transform of the solution y(t) to the initial value problem below. y" +9y = 5t²-6, y(0) = 0, y'(0) = -8 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. ... 64 5t² 64 8 sin 3t Y(s) = + 81 9 81 3 + cos 3t
Solve for Y(s), the Laplace transform of the solution y(t) to the initial value problem below. y" +9y = 5t²-6, y(0) = 0, y'(0) = -8 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. ... 64 5t² 64 8 sin 3t Y(s) = + 81 9 81 3 + cos 3t
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Help solve my answer is wrong
![**Solving Initial Value Problems using Laplace Transforms**
**Problem Statement:**
Solve for \( Y(s) \), the Laplace transform of the solution \( y(t) \), to the initial value problem below:
\[ y'' + 9y = 5t^2 - 6, \; y(0) = 0, \; y'(0) = -8 \]
**Useful Resources:**
Click here to view the table of Laplace transforms.
Click here to view the table of properties of Laplace transforms.
---
### Solution:
\[ Y(s) = -\frac{64}{81s} + \frac{5t^2}{9} + \frac{64}{81} \cos 3t - \frac{8 \sin 3t}{3} \]
Explanation of steps:
1. **Setting up the Laplace transform of the differential equation:** We start with the given differential equation and take the Laplace transform of both sides.
2. **Applying initial conditions:** We utilize the initial conditions \( y(0) = 0 \) and \( y'(0) = -8 \) in the Laplace transform domain.
3. **Solving for \( Y(s) \):** We manipulate the resulting algebraic equation to solve for the Laplace transform \( Y(s) \) of the function \( y(t) \).
4. **Inverse Laplace Transform:** Finally, we use the inverse Laplace transform to convert \( Y(s) \) back to \( y(t) \), if required.
By following these steps, you can find the solution to initial value problems using Laplace transforms and understand the behavior of the function \( y(t) \) over time.
---
Figure Explanation:
- The equation under "Solution" provides the Laplace transform \( Y(s) \) in terms of the variable \( s \).
- The terms in the solution include both exponential functions of \( s \) and trigonometric functions like cosine and sine, indicating the influence of the initial conditions and the non-homogeneous part of the original differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbba0909-7b4f-4e80-8e24-06dfda2d061e%2Faefaae20-7f1b-4419-ad46-8487a4c4e2e1%2Frqvfl5l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving Initial Value Problems using Laplace Transforms**
**Problem Statement:**
Solve for \( Y(s) \), the Laplace transform of the solution \( y(t) \), to the initial value problem below:
\[ y'' + 9y = 5t^2 - 6, \; y(0) = 0, \; y'(0) = -8 \]
**Useful Resources:**
Click here to view the table of Laplace transforms.
Click here to view the table of properties of Laplace transforms.
---
### Solution:
\[ Y(s) = -\frac{64}{81s} + \frac{5t^2}{9} + \frac{64}{81} \cos 3t - \frac{8 \sin 3t}{3} \]
Explanation of steps:
1. **Setting up the Laplace transform of the differential equation:** We start with the given differential equation and take the Laplace transform of both sides.
2. **Applying initial conditions:** We utilize the initial conditions \( y(0) = 0 \) and \( y'(0) = -8 \) in the Laplace transform domain.
3. **Solving for \( Y(s) \):** We manipulate the resulting algebraic equation to solve for the Laplace transform \( Y(s) \) of the function \( y(t) \).
4. **Inverse Laplace Transform:** Finally, we use the inverse Laplace transform to convert \( Y(s) \) back to \( y(t) \), if required.
By following these steps, you can find the solution to initial value problems using Laplace transforms and understand the behavior of the function \( y(t) \) over time.
---
Figure Explanation:
- The equation under "Solution" provides the Laplace transform \( Y(s) \) in terms of the variable \( s \).
- The terms in the solution include both exponential functions of \( s \) and trigonometric functions like cosine and sine, indicating the influence of the initial conditions and the non-homogeneous part of the original differential equation.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
