Solve y""-2y"-5y'+6y=-12xe +2e¹; y(0)=2, y(0)=7, y"(0)=9 using the method of aplace transforms. Feel free to use the appropriate table and formula.
Solve y""-2y"-5y'+6y=-12xe +2e¹; y(0)=2, y(0)=7, y"(0)=9 using the method of aplace transforms. Feel free to use the appropriate table and formula.
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 Higher-Order Differential Equations Using Laplace Transforms
In this lesson, we will solve a fourth-order differential equation using the method of Laplace transforms. The given differential equation is:
\[ y^{(4)} - 2y''' - 5y'' + 6y = -12xe^t + 2e^t \]
We are provided with the initial conditions:
\[
y(0) = 2, \quad y'(0) = 7, \quad y''(0) = 9
\]
### Steps to Solve Using Laplace Transforms
1. **Take the Laplace Transform of Both Sides:**
Apply the Laplace transform to each term in the differential equation. Recall that the Laplace transform of a derivative is given by:
\[
L\{y^{(n)}(t)\} = s^n Y(s) - s^{n-1} y(0) - s^{n-2} y'(0) - \cdots - y^{(n-1)}(0)
\]
Hence, the Laplace transform of the given differential equation is:
\[
L\{y^{(4)}\} - 2L\{y^{(3)}\} - 5L\{y''\} + 6L\{y\} = L\{-12xe^t + 2e^t\}
\]
2. **Substitute Initial Conditions:**
Replace the initial conditions \( y(0) = 2 \), \( y'(0) = 7 \), and \( y''(0) = 9 \) into the transformed equation.
3. **Solve for \( Y(s) \):**
Isolate \( Y(s) \) to express it in terms of \( s \) and the given functions on the right-hand side. This will typically involve algebraic manipulation.
4. **Inverse Laplace Transform:**
Finally, take the inverse Laplace transform of \( Y(s) \) to return to the time domain and find the solution \( y(t) \).
### Graphs and Diagrams
This problem does not include any graphs or diagrams. However, visualizations such as plot points or steps in solving differential equations can greatly aid in understanding. When working through these transformations and solutions, a graph plotting the general](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F946e5b6f-098d-47d1-a6b1-15872ebd1054%2F2fef9355-6ac5-4b8a-b9b5-9fb61f05468d%2Fd3lkv58hd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Higher-Order Differential Equations Using Laplace Transforms
In this lesson, we will solve a fourth-order differential equation using the method of Laplace transforms. The given differential equation is:
\[ y^{(4)} - 2y''' - 5y'' + 6y = -12xe^t + 2e^t \]
We are provided with the initial conditions:
\[
y(0) = 2, \quad y'(0) = 7, \quad y''(0) = 9
\]
### Steps to Solve Using Laplace Transforms
1. **Take the Laplace Transform of Both Sides:**
Apply the Laplace transform to each term in the differential equation. Recall that the Laplace transform of a derivative is given by:
\[
L\{y^{(n)}(t)\} = s^n Y(s) - s^{n-1} y(0) - s^{n-2} y'(0) - \cdots - y^{(n-1)}(0)
\]
Hence, the Laplace transform of the given differential equation is:
\[
L\{y^{(4)}\} - 2L\{y^{(3)}\} - 5L\{y''\} + 6L\{y\} = L\{-12xe^t + 2e^t\}
\]
2. **Substitute Initial Conditions:**
Replace the initial conditions \( y(0) = 2 \), \( y'(0) = 7 \), and \( y''(0) = 9 \) into the transformed equation.
3. **Solve for \( Y(s) \):**
Isolate \( Y(s) \) to express it in terms of \( s \) and the given functions on the right-hand side. This will typically involve algebraic manipulation.
4. **Inverse Laplace Transform:**
Finally, take the inverse Laplace transform of \( Y(s) \) to return to the time domain and find the solution \( y(t) \).
### Graphs and Diagrams
This problem does not include any graphs or diagrams. However, visualizations such as plot points or steps in solving differential equations can greatly aid in understanding. When working through these transformations and solutions, a graph plotting the general
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