Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Consider the following initial value problem.

\[ y'' + 10y' + 41y = \delta(t - \pi) + \delta(t - 7\pi), \quad y(0) = 1, \quad y'(0) = 0 \]

Find the Laplace transform of the differential equation. (Write your answer as a function of \( s \)).

\[ \mathcal{L}\{y\} = \boxed{\frac{1}{4} e^{-5t} \sin(4t)} \, \, \textcolor{red}{\textit{X}} \]

Use the Laplace transform to solve the given initial-value problem.

\[ y(t) = \left( \boxed{e^{-5t} \cos (4t)} \, \, \textcolor{red}{\textit{X}} \right) + \left( \boxed{ \frac{5}{4} e^{-5t} \sin(4t) + \frac{1}{4} e^{-5t} + 5\pi \sin(4t)} \right) \cdot u(t - \pi) + \left( \boxed{-\frac{1}{4} e^{-5t} + 35\pi \sin(4t)} \right) \cdot u(t - 7\pi) \, \, \textcolor{green}{\checkmark} \)

In the image, there are several boxed equations and expressions. The boxed equations followed by a red cross (✗) indicate incorrect solutions, while the boxed equation followed by a green checkmark (✓) indicates a correct solution. 

**Explanation of Incorrect Solutions**

1. \( \mathcal{L}\{y\} = \frac{1}{4} e^{-5t} \sin(4t) \, \, \textcolor{red}{\textit{X}} \): This equation contains errors in the computation of the Laplace transform of the differential equation.

2. 

\[ y(t) = \left( e^{-5t} \cos (4t) \, \, \textcolor{red}{\textit{X}} \right) + \left( \frac{5}{4} e^{-5t} \sin(4t) + \frac{1}{4} e^{-5
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Transcribed Image Text:Consider the following initial value problem. \[ y'' + 10y' + 41y = \delta(t - \pi) + \delta(t - 7\pi), \quad y(0) = 1, \quad y'(0) = 0 \] Find the Laplace transform of the differential equation. (Write your answer as a function of \( s \)). \[ \mathcal{L}\{y\} = \boxed{\frac{1}{4} e^{-5t} \sin(4t)} \, \, \textcolor{red}{\textit{X}} \] Use the Laplace transform to solve the given initial-value problem. \[ y(t) = \left( \boxed{e^{-5t} \cos (4t)} \, \, \textcolor{red}{\textit{X}} \right) + \left( \boxed{ \frac{5}{4} e^{-5t} \sin(4t) + \frac{1}{4} e^{-5t} + 5\pi \sin(4t)} \right) \cdot u(t - \pi) + \left( \boxed{-\frac{1}{4} e^{-5t} + 35\pi \sin(4t)} \right) \cdot u(t - 7\pi) \, \, \textcolor{green}{\checkmark} \) In the image, there are several boxed equations and expressions. The boxed equations followed by a red cross (✗) indicate incorrect solutions, while the boxed equation followed by a green checkmark (✓) indicates a correct solution. **Explanation of Incorrect Solutions** 1. \( \mathcal{L}\{y\} = \frac{1}{4} e^{-5t} \sin(4t) \, \, \textcolor{red}{\textit{X}} \): This equation contains errors in the computation of the Laplace transform of the differential equation. 2. \[ y(t) = \left( e^{-5t} \cos (4t) \, \, \textcolor{red}{\textit{X}} \right) + \left( \frac{5}{4} e^{-5t} \sin(4t) + \frac{1}{4} e^{-5
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Follow-up Question

Please help with the boxes marked red. Previous Bartleby answer was incorrect

Consider the following initial value problem.
-
y" + 10y' + 41y = (t = π) + 8(t - 7π), y(0) = 1, y'(0) = 0
Find the Laplace transform of the differential equation. (Write your answer as a function of s.)
-5tsin (4t)
L{y}
=
Use the Laplace transform to solve the given initial-value problem.
) + ( ½ e
y(t) = (
e-5tcos (4t) +
-5t sin (4t)
5 -5t
е
4e
¸-5(t−7ñ) sin(4(t− 7ñ))
). u(t = π) +
¹⁄e¯5(t−ñ) sin(4(t−x))__). 2(t-
4
7π
expand button
Transcribed Image Text:Consider the following initial value problem. - y" + 10y' + 41y = (t = π) + 8(t - 7π), y(0) = 1, y'(0) = 0 Find the Laplace transform of the differential equation. (Write your answer as a function of s.) -5tsin (4t) L{y} = Use the Laplace transform to solve the given initial-value problem. ) + ( ½ e y(t) = ( e-5tcos (4t) + -5t sin (4t) 5 -5t е 4e ¸-5(t−7ñ) sin(4(t− 7ñ)) ). u(t = π) + ¹⁄e¯5(t−ñ) sin(4(t−x))__). 2(t- 4 7π
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Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

Please help with the boxes marked red. Previous Bartleby answer was incorrect

Consider the following initial value problem.
-
y" + 10y' + 41y = (t = π) + 8(t - 7π), y(0) = 1, y'(0) = 0
Find the Laplace transform of the differential equation. (Write your answer as a function of s.)
-5tsin (4t)
L{y}
=
Use the Laplace transform to solve the given initial-value problem.
) + ( ½ e
y(t) = (
e-5tcos (4t) +
-5t sin (4t)
5 -5t
е
4e
¸-5(t−7ñ) sin(4(t− 7ñ))
). u(t = π) +
¹⁄e¯5(t−ñ) sin(4(t−x))__). 2(t-
4
7π
expand button
Transcribed Image Text:Consider the following initial value problem. - y" + 10y' + 41y = (t = π) + 8(t - 7π), y(0) = 1, y'(0) = 0 Find the Laplace transform of the differential equation. (Write your answer as a function of s.) -5tsin (4t) L{y} = Use the Laplace transform to solve the given initial-value problem. ) + ( ½ e y(t) = ( e-5tcos (4t) + -5t sin (4t) 5 -5t е 4e ¸-5(t−7ñ) sin(4(t− 7ñ)) ). u(t = π) + ¹⁄e¯5(t−ñ) sin(4(t−x))__). 2(t- 4 7π
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