8y" +45y" 128y' - 720y = 5e-5t y(0) = 0 y' (0) = 0 y" (0) = 1 into an algebraic equation by taking the Laplace transform of each side. Use y for the Laplace transform of y. (not Y(s)). Therefore Y = 34+ Taking the inverse Laplace transform we get y = 1 $44 + = $+5
8y" +45y" 128y' - 720y = 5e-5t y(0) = 0 y' (0) = 0 y" (0) = 1 into an algebraic equation by taking the Laplace transform of each side. Use y for the Laplace transform of y. (not Y(s)). Therefore Y = 34+ Taking the inverse Laplace transform we get y = 1 $44 + = $+5
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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transform the differenial equation
please follow the question and fill the gaps
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![### Transcription for Educational Website
---
**Given Differential Equation:**
\[ 8y''' + 45y'' - 128y' - 720y = 5e^{-5t} \]
**Initial Conditions:**
\[
\begin{align*}
y(0) &= 0 \\
y'(0) &= 0 \\
y''(0) &= 1
\end{align*}
\]
**Step 1: Transform the differential equation into an algebraic equation by taking the Laplace transform of each side. Use \( Y \) for the Laplace transform of \( y \), not \( Y(s) \).**
\[ 8Y''' + 45Y'' - 128Y' - 720Y = \frac{5}{s+5} \]
**Step 2: Solve for \( Y \)**
Therefore,
\[ Y = \frac{1}{s-4} + \frac{1}{s+4} + \frac{1}{s+5} \]
**Step 3: Taking the inverse Laplace transform we get**
\[ y = e^{4t} + e^{-4t} + e^{-5t} \]
---
By following these steps, you can transform a differential equation into an algebraic equation using the Laplace transform, solve for the transformed variable, and then take the inverse Laplace transform to find the solution in the time domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ebcf202-e71c-4e7a-b7be-45ee67012454%2Ffb42b47b-9ac1-4dee-a629-de8aa8b3d1fb%2F9icce5p_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Website
---
**Given Differential Equation:**
\[ 8y''' + 45y'' - 128y' - 720y = 5e^{-5t} \]
**Initial Conditions:**
\[
\begin{align*}
y(0) &= 0 \\
y'(0) &= 0 \\
y''(0) &= 1
\end{align*}
\]
**Step 1: Transform the differential equation into an algebraic equation by taking the Laplace transform of each side. Use \( Y \) for the Laplace transform of \( y \), not \( Y(s) \).**
\[ 8Y''' + 45Y'' - 128Y' - 720Y = \frac{5}{s+5} \]
**Step 2: Solve for \( Y \)**
Therefore,
\[ Y = \frac{1}{s-4} + \frac{1}{s+4} + \frac{1}{s+5} \]
**Step 3: Taking the inverse Laplace transform we get**
\[ y = e^{4t} + e^{-4t} + e^{-5t} \]
---
By following these steps, you can transform a differential equation into an algebraic equation using the Laplace transform, solve for the transformed variable, and then take the inverse Laplace transform to find the solution in the time domain.
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