Solve for Y(s), the Laplace transform of the solution y(t) to the initial value problem y" +3y=t²-1. y(0) = 0, y'(0) = -6 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. ... Y(s) =
Solve for Y(s), the Laplace transform of the solution y(t) to the initial value problem y" +3y=t²-1. y(0) = 0, y'(0) = -6 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. ... Y(s) =
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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Help with Laplace transform question
![**Solve for Y(s): The Laplace Transform of the Solution y(t) for the Initial Value Problem**
Given the initial value problem:
\[ y'' + 3y = t^2 - 1, \quad y(0) = 0, \quad y'(0) = -6 \]
Your task is to find \( Y(s) \), the Laplace transform of the solution \( y(t) \), by following these steps:
1. **Laplace Transform of the Differential Equation**:
Apply the Laplace transform to both sides of the given differential equation.
2. **Incorporate Initial Conditions**:
Use the given initial conditions to solve for \( Y(s) \).
\[
\mathcal{L}\{y''\} + 3\mathcal{L}\{y\} = \mathcal{L}\{t^2 - 1\}
\]
\[
\Rightarrow s^2Y(s) - sy(0) - y'(0) + 3Y(s) = \frac{2}{s^3} - \frac{1}{s}
\]
\[
\Rightarrow s^2Y(s) + 6 + 3Y(s) = \frac{2}{s^3} - \frac{1}{s}
\]
Simplify and solve for \( Y(s) \).
3. **Resources for Solving**:
* [Click here to view the table of Laplace transforms](#)
* [Click here to view the table of properties of Laplace transforms](#)
Finally, input your solution in the provided box.
\[ \boxed{ \quad } Y(s) \boxed{\quad} \]
This framework should help you systematically approach solving the Laplace transform for the given differential equation using the provided initial conditions. Good luck!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbba0909-7b4f-4e80-8e24-06dfda2d061e%2Fb6e90752-a239-484f-8f25-190493f9e45a%2Fhtgzsl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solve for Y(s): The Laplace Transform of the Solution y(t) for the Initial Value Problem**
Given the initial value problem:
\[ y'' + 3y = t^2 - 1, \quad y(0) = 0, \quad y'(0) = -6 \]
Your task is to find \( Y(s) \), the Laplace transform of the solution \( y(t) \), by following these steps:
1. **Laplace Transform of the Differential Equation**:
Apply the Laplace transform to both sides of the given differential equation.
2. **Incorporate Initial Conditions**:
Use the given initial conditions to solve for \( Y(s) \).
\[
\mathcal{L}\{y''\} + 3\mathcal{L}\{y\} = \mathcal{L}\{t^2 - 1\}
\]
\[
\Rightarrow s^2Y(s) - sy(0) - y'(0) + 3Y(s) = \frac{2}{s^3} - \frac{1}{s}
\]
\[
\Rightarrow s^2Y(s) + 6 + 3Y(s) = \frac{2}{s^3} - \frac{1}{s}
\]
Simplify and solve for \( Y(s) \).
3. **Resources for Solving**:
* [Click here to view the table of Laplace transforms](#)
* [Click here to view the table of properties of Laplace transforms](#)
Finally, input your solution in the provided box.
\[ \boxed{ \quad } Y(s) \boxed{\quad} \]
This framework should help you systematically approach solving the Laplace transform for the given differential equation using the provided initial conditions. Good luck!
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