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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Consider the following initial value problem. y" + 10y' + 41y = 8(t) + 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.) L{y} = -5t sin (4t) 1 4 X Use the Laplace transform to solve the given initial-value problem. y(t) = e-5t cos (4t) X )+(5e-5sin (41) + 4 1-5t+5 sin (4t) 4 X ).u(t− n) + ( 1 Te -5t+35 sin (4t) X ). 2(t- t - 7π

Consider the following initial value problem. y" + 10y' + 41y = 8(t) + 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.) L{y} = -5t sin (4t) 1 4 X Use the Laplace transform to solve the given initial-value problem. y(t) = e-5t cos (4t) X )+(5e-5sin (41) + 4 1-5t+5 sin (4t) 4 X ).u(t− n) + ( 1 Te -5t+35 sin (4t) X ). 2(t- t - 7π

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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Question

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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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
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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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