Solve using the method of Laplace transforms /" – 2y' – 84 = 0 y (0) y (0) %3D 3.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![### Solving Differential Equations Using Laplace Transforms
In this example, we will solve a second-order linear differential equation using the method of Laplace transforms. The differential equation and initial conditions are given as follows:
Differential Equation:
\[ y'' - 2y' - 8y = 0 \]
Initial Conditions:
\[ y(0) = 3 \]
\[ y'(0) = 6 \]
#### Steps to Solve Using Laplace Transforms:
1. **Apply the Laplace Transform to both sides of the differential equation:**
Recall that the Laplace transform of \( y(t) \) is \( Y(s) \). Using the properties of the Laplace transform, we have:
\[ \mathcal{L}\{y''(t)\} = s^2Y(s) - sy(0) - y'(0) \]
\[ \mathcal{L}\{y'(t)\} = sY(s) - y(0) \]
\[ \mathcal{L}\{y(t)\} = Y(s) \]
Substituting \( y''(t) \), \( y'(t) \), and \( y(t) \) into the differential equation gives:
\[ s^2Y(s) - sy(0) - y'(0) - 2(sY(s) - y(0)) - 8Y(s) = 0 \]
2. **Substitute the initial conditions into the transformed equation:**
Using \( y(0) = 3 \) and \( y'(0) = 6 \):
\[ s^2Y(s) - 3s - 6 - 2(sY(s) - 3) - 8Y(s) = 0 \]
\[ s^2Y(s) - 3s - 6 - 2sY(s) + 6 - 8Y(s) = 0 \]
\[ (s^2 - 2s - 8)Y(s) - 3s - 6 + 6 = 0 \]
\[ (s^2 - 2s - 8)Y(s) = 3s \]
3. **Solve for \( Y(s) \):**
\[ Y(s) = \frac{3s}{s^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20e00ab3-e90c-493d-a4f6-cdfa310ac890%2Fc478fc11-76fb-4e0a-87b9-a47ce95b30ad%2F6p86nof_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Differential Equations Using Laplace Transforms
In this example, we will solve a second-order linear differential equation using the method of Laplace transforms. The differential equation and initial conditions are given as follows:
Differential Equation:
\[ y'' - 2y' - 8y = 0 \]
Initial Conditions:
\[ y(0) = 3 \]
\[ y'(0) = 6 \]
#### Steps to Solve Using Laplace Transforms:
1. **Apply the Laplace Transform to both sides of the differential equation:**
Recall that the Laplace transform of \( y(t) \) is \( Y(s) \). Using the properties of the Laplace transform, we have:
\[ \mathcal{L}\{y''(t)\} = s^2Y(s) - sy(0) - y'(0) \]
\[ \mathcal{L}\{y'(t)\} = sY(s) - y(0) \]
\[ \mathcal{L}\{y(t)\} = Y(s) \]
Substituting \( y''(t) \), \( y'(t) \), and \( y(t) \) into the differential equation gives:
\[ s^2Y(s) - sy(0) - y'(0) - 2(sY(s) - y(0)) - 8Y(s) = 0 \]
2. **Substitute the initial conditions into the transformed equation:**
Using \( y(0) = 3 \) and \( y'(0) = 6 \):
\[ s^2Y(s) - 3s - 6 - 2(sY(s) - 3) - 8Y(s) = 0 \]
\[ s^2Y(s) - 3s - 6 - 2sY(s) + 6 - 8Y(s) = 0 \]
\[ (s^2 - 2s - 8)Y(s) - 3s - 6 + 6 = 0 \]
\[ (s^2 - 2s - 8)Y(s) = 3s \]
3. **Solve for \( Y(s) \):**
\[ Y(s) = \frac{3s}{s^
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