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
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 59E: According to the solution in Exercise 58 of the differential equation for Newtons law of cooling,...
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96023f4d-6281-4c81-803e-adfe3e4197d6%2F1aa3e114-aab5-4bac-94a3-18043d90c00a%2Fvx32xfk_processed.png&w=3840&q=75)
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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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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