Explanation Formula used: The Laplace transformation of f : “The Laplace transform of f which is dented by L { f ( t ) } = F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t ”. Calculation: The given function is f ( t ) = { t , 0 ≤ t < 1 2 − t , 1 ≤ t < 2 0 , 2 ≤ t < ∞ } . Compute the Laplace transformation of f ( t ) = { t , 0 ≤ t < 1 2 − t , 1 ≤ t < 2 0 , 2 ≤ t < ∞ } as follows. Substitute f ( t ) = t for 0 ≤ t < 1 , f ( t ) = 2 − t for 1 ≤ t < 2 , and f ( t ) = 0 for 2 ≤ t < ∞ in F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t and evaluate further. F ( s ) = ∫ 0 1 e − s t ( t ) d t + ∫ 1 2 e − s t ( 2 − t ) d t + ∫ 2 ∞ e − s t ( 0 ) d t F ( s ) = ∫ 0 1 t e − s t d t + ∫ 1 2 ( 2 − t ) e − s t d t Let F ( s ) = ∫ 0 1 t e − s t d t + ∫ 1 2 ( 2 − t ) e − s t d t be equation (1). First compute the value of ∫ 0 1 t e − s t d t as follows. Here, u = t , then d u = d t and d v = e − s t d t . That implies, ∫ d v = ∫ e − s t d t v = − 1 s ∫ e u d u [ let u = − s t , then d u = − s d t ] = − 1 s [ e u ] = − e − s t s The equation (1) becomes, ∫ 0 1 t e − s t d t = [ t ⋅ − e − s t s ] 0 1 − ∫ 0 1 − e − s t s d t = [ 1 ⋅ − e − s s − 0 ] + 1 s ∫ 0 1 e − s t d t = − e − s s + 1 s ∫ 0 1 − 1 s e w d w [ let w = − s t , then d w = − s d t ] = − e − s s − 1 s 2 [ e w ] 0 1 On further simplification, ∫ 0 1 t e − s t d t = − e − s s − 1 s 2 [ e − s t ] 0 1 [ ∵ w = − s t ] = − e − s s − 1 s 2 [ e − s − 1 ] = − e − s s − e − s s 2 + 1 s 2 = 1 − s e − s − e − s s 2 = 1 − ( s + 1 ) e − s s 2 Now compute the value of ∫ 1 2 ( 2 − t ) e − s t d t as follows

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

See similar textbooks

Question

Chapter 6.1, Problem 18P

To determine

The Laplace transformation of the given function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.1, Problem 17P

Chapter 6.1, Problem 19P

Students have asked these similar questions

3. Consider the polynomial equation 6-iz+7z² - iz³ +z = 0 for which the roots are 3i, -2i, -i, and i. (a) Verify the relations between this roots and the coefficients of the polynomial. (b) Find the annulus region in which the roots lie.

The managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c. Question content area bottom Part 1 a. Develop a simple linear regression model between billable hours and overhead costs. Overhead Costsequals=212495.2212495.2plus+left parenthesis 42.4857 right parenthesis42.485742.4857times×Billable Hours (Round the constant to one decimal place as needed. Round the coefficient to four decimal places as needed. Do not include the $ symbol in your answers.) Part 2 b. Interpret the coefficients of your regression model. Specifically, what does the fixed component of the model mean to the consulting firm? Interpret the fixed term, b 0b0, if appropriate. Choose the correct answer below. A. The value of b 0b0 is the predicted billable hours for an overhead cost of 0 dollars. B. It is not appropriate to interpret b 0b0, because its value…

3. Consider the polynomial equation 6-iz+7z2-iz³ +z = 0 for which the roots are 3i, -2i, -i, and i. (a) Verify the relations between this roots and the coefficients of the polynomial. (b) Find the annulus region in which the roots lie.

Chapter 6 Solutions

Elementary Differential Equations and Boundary Value Problems, Enhanced

Ch. 6.1 - Prob. 1P Ch. 6.1 - Prob. 2P Ch. 6.1 - Prob. 3P Ch. 6.1 - Prob. 4P Ch. 6.1 - Prob. 5P Ch. 6.1 - Prob. 6P Ch. 6.1 - Prob. 7P Ch. 6.1 - Prob. 8P Ch. 6.1 - Prob. 9P Ch. 6.1 - Prob. 10P

Ch. 6.1 - Prob. 11P Ch. 6.1 - Prob. 12P Ch. 6.1 - Prob. 13P Ch. 6.1 - Prob. 14P Ch. 6.1 - Prob. 15P Ch. 6.1 - Prob. 16P Ch. 6.1 - Prob. 17P Ch. 6.1 - Prob. 18P Ch. 6.1 - Prob. 19P Ch. 6.1 - Prob. 20P Ch. 6.1 - Prob. 21P Ch. 6.1 - Prob. 22P Ch. 6.1 - Prob. 23P Ch. 6.1 - Prob. 24P Ch. 6.2 - Prob. 1P Ch. 6.2 - Prob. 2P Ch. 6.2 - Prob. 3P Ch. 6.2 - Prob. 4P Ch. 6.2 - Prob. 5P Ch. 6.2 - Prob. 6P Ch. 6.2 - Prob. 7P Ch. 6.2 - Prob. 8P Ch. 6.2 - Prob. 9P Ch. 6.2 - Prob. 10P Ch. 6.2 - Prob. 11P Ch. 6.2 - Prob. 12P Ch. 6.2 - Prob. 13P Ch. 6.2 - Prob. 14P Ch. 6.2 - Prob. 15P Ch. 6.2 - Prob. 16P Ch. 6.2 - Prob. 17P Ch. 6.2 - Prob. 18P Ch. 6.2 - Prob. 19P Ch. 6.2 - Prob. 20P Ch. 6.2 - Prob. 21P Ch. 6.2 - Prob. 22P Ch. 6.2 - Prob. 23P Ch. 6.2 - Prob. 24P Ch. 6.2 - Prob. 25P Ch. 6.2 - Prob. 26P Ch. 6.2 - Prob. 27P Ch. 6.2 - Prob. 28P Ch. 6.2 - Prob. 29P Ch. 6.3 - Prob. 1P Ch. 6.3 - Prob. 2P Ch. 6.3 - Prob. 3P Ch. 6.3 - Prob. 4P Ch. 6.3 - Prob. 5P Ch. 6.3 - Prob. 6P Ch. 6.3 - Prob. 7P Ch. 6.3 - Prob. 8P Ch. 6.3 - Prob. 9P Ch. 6.3 - Prob. 10P Ch. 6.3 - Prob. 11P Ch. 6.3 - Prob. 12P Ch. 6.3 - Prob. 13P Ch. 6.3 - Prob. 14P Ch. 6.3 - Prob. 15P Ch. 6.3 - Prob. 16P Ch. 6.3 - Prob. 17P Ch. 6.3 - Prob. 18P Ch. 6.3 - Prob. 19P Ch. 6.3 - Prob. 20P Ch. 6.3 - Prob. 21P Ch. 6.3 - Prob. 22P Ch. 6.3 - Prob. 23P Ch. 6.3 - Prob. 24P Ch. 6.3 - Prob. 25P Ch. 6.3 - Prob. 26P Ch. 6.3 - Prob. 27P Ch. 6.3 - Prob. 28P Ch. 6.3 - Prob. 29P Ch. 6.3 - Prob. 30P Ch. 6.4 - Prob. 1P Ch. 6.4 - Prob. 2P Ch. 6.4 - Prob. 3P Ch. 6.4 - Prob. 4P Ch. 6.4 - Prob. 5P Ch. 6.4 - Prob. 6P Ch. 6.4 - Prob. 7P Ch. 6.4 - In each of Problems 1 through 8: a. Sketch the...Ch. 6.4 - Prob. 9P Ch. 6.4 - Prob. 10P Ch. 6.4 - Prob. 11P Ch. 6.4 - Prob. 12P Ch. 6.4 - Prob. 13P Ch. 6.4 - Prob. 14P Ch. 6.4 - Prob. 15P Ch. 6.4 - Prob. 16P Ch. 6.4 - Prob. 17P Ch. 6.5 - Prob. 1P Ch. 6.5 - Prob. 2P Ch. 6.5 - Prob. 3P Ch. 6.5 - Prob. 4P Ch. 6.5 - Prob. 5P Ch. 6.5 - Prob. 6P Ch. 6.5 - Prob. 7P Ch. 6.5 - Prob. 8P Ch. 6.5 - Prob. 9P Ch. 6.5 - Prob. 10P Ch. 6.5 - Prob. 11P Ch. 6.5 - Prob. 12P Ch. 6.5 - Prob. 13P Ch. 6.5 - Prob. 14P Ch. 6.5 - Prob. 15P Ch. 6.5 - Prob. 16P Ch. 6.5 - Prob. 17P Ch. 6.5 - Prob. 18P Ch. 6.5 - Prob. 19P Ch. 6.6 - Prob. 1P Ch. 6.6 - Prob. 2P Ch. 6.6 - Prob. 3P Ch. 6.6 - Prob. 4P Ch. 6.6 - Prob. 5P Ch. 6.6 - In each of Problems 4 through 7, find the Laplace...Ch. 6.6 - Prob. 7P Ch. 6.6 - Prob. 8P Ch. 6.6 - Prob. 9P Ch. 6.6 - Prob. 10P Ch. 6.6 - Prob. 11P Ch. 6.6 - Prob. 12P Ch. 6.6 - Prob. 13P Ch. 6.6 - Prob. 14P Ch. 6.6 - Prob. 15P Ch. 6.6 - Prob. 16P Ch. 6.6 - Prob. 17P Ch. 6.6 - Prob. 18P Ch. 6.6 - Prob. 19P Ch. 6.6 - Prob. 20P Ch. 6.6 - Prob. 21P

Knowledge Booster