Explanation Given: The initial value problem is w ″ + 6 w ′ + 5 w = e t δ ( t − 1 ) , w ( 0 ) = 0 , w ′ ( 0 ) = 4 . Approach: a) Write the initial value problem. b) Apply the laplace transformation on both side. c) Apply linearity property of laplace transforms as L { a + b } = L { a } + L { b } . d) Take L { w } = W ( s ) . e) Find derivative of laplace transform, L { w ″ } = s 2 W ( s ) − s w ( 0 ) − w ′ ( 0 ) L { w ′ } = s W ( s ) − w ( 0 ) f) Apply partial fraction to simplify the terms. g) Write laplace transform of Dirac delta function as L { e t δ ( t − a ) } ( s ) = ∫ 0 ∞ e t ⋅ e − a st δ ( t − a ) d t , and δ ( t − 1 ) = 1 when t = 1 . Also use property, L { e t } ( s ) = 1 s − 1 L − 1 { 1 s − a } = e at h) Apply inverse laplace transform on both side. i) Use second translation theorem property in inverse laplace transforms as L − 1 { e − as F ( s ) } ( t ) = f ( t − a ) u ( t − a ) L − 1 { 1 s − a } = e at L − 1 { a ⋅ f ( s ) + b ⋅ g ( s ) } = a ⋅ L − 1 { f ( s ) } + b ⋅ L − 1 { g ( s ) } L − 1 { f ( t ) } = a ⋅ L − 1 { f ( t ) } , Calculation: Write the initial value problem, w ″ + 6 w ′ + 5 w = e t δ ( t − 1 ) . Apply the laplace transformation on both side, Then, L { w ″ + 6 w ′ + 5 w } = L { e t δ ( t − 1 ) } Apply linearity property of laplace transforms as, L { w ″ } − 6 L { w ′ } + 5 L { w } = L { e t δ ( t − 1 ) } ... ( 1 ) Find derivative of laplace transform. L { w ″ } = s 2 W ( s ) − s w ( 0 ) − w ′ ( 0 ) L { w ′ } = s W ( s ) − w ( 0 ) Take L { w } = W ( s ) . Write laplace transform of Dirac delta function as, L { e t δ ( t − 1 ) } ( s ) = ∫ 0 ∞ e t ⋅ e − st δ ( t − 1 ) d t Then, L { e t δ ( t − 1 ) } ( s ) = ∫ 0 ∞ e t-st δ ( t − 1 ) d t Because, δ ( t − 1 ) = 1 when t = 1 . So, L { e t δ ( t − 1 ) } ( s ) = e t ( 1-s ) = e ( 1-s ) . Substitute values in equation ( 1 ) . Then equation ( 1 ) become, s 2 W ( s ) − s w ( 0 ) − w ′ ( 0 ) + 6 ( s W ( s ) − w ( 0 ) ) + 5 W ( s ) = e ( 1-s ) W ( s ) ( s 2 + 6 s + 5 ) = e ( 1-s ) + 4 W ( s ) = e ( 1-s ) ( s 2 + 6 s + 5 ) + 4 ( s 2 + 6 s + 5 ) , Convert into simple form,

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

See similar textbooks

Videos

Question

Chapter 7.9, Problem 19E

To determine

To find:

The solution of initial value problem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.9, Problem 18E

Chapter 7.9, Problem 20E

Students have asked these similar questions

CClever Portal CA Co Managed bookmarks ixl.com/math/geometry/surface-area-and-volume-review?ImsPlatform=1 IXL My IXL Learning Assessment Geometry > U.15 Surface area and volume review TER C Learn with an example or ► Watch a video What is the surface area of this cone? Use л ≈ 3.14 and round your answer to the nearest hundredth. 7 ft 8 ft Submit square feet Type here to search Work it out Not feeling ready yet? These can help: 0 F1 1 2 F2 F3 #3 X F4 C F5 $ 4 % L E F6 + 回 F7 F8 F9 & об г

In Gallup's Annual Consumption Habits Poll, telephone interviews were conducted for a random sample of 1014 adults aged 18 and over. One of the questions was, "How many cups of coffee, if any, do you drink on an average day?" The following table shows the results obtained (Gallup website, August 6, 2012). Number of Cups per Day 0 1 2 3 4 or more Number of Responses 365 264 193 91 101 Define a random variable x = number of cups of coffee consumed on an average day. Let x=4 represent four or more cups. a. Develop a probability distribution for x. b. Compute the expected value of x. c. Compute the variance of x. d. Suppose we are only interested in adults who drink at least one cup of coffee on an average day. For this group, let y = the number of cups of coffee consumed on an average day. Compute the expected value of y and compare it to the expected value of x.

Ten major college football bowl games were played in January 2010, with the University of Alabama beating the University of Texas 37 to 21 to become the national champion of college football. The results of the 10 bowl games follow (USA Today, January 8, 2010). Bowl Game Score Outback Gator Auburn 38 Northwestern 35 Florida State 33 West Virginia 21 Capital One Penn State 19 LSU 17 Rose Ohio State 26 Oregon 17 Florida 51 Cincinnati 24 Sugar Cotton Alamo Fiesta Mississippi State 21 Oklahoma State 7 Texas Tech 41 Michigan State 31 Boise State 17 TCU 10 Orange Iowa 24 Georgia Tech 14 Championship Alabama 37 Texas 21 Predicted Point Margin Actual Point Margin 5 3 1 12 3 -2 14 3 9 -4 -3 4 229740706 14 10 10 16 The predicted winning point margin was based on Las Vegas betting odds approximately one week before the bowl games were played. For example, Auburn was predicted to beat Northwestern in the Outback Bowl by five points. The actual winning point margin for Au- burn was three points. A…

Chapter 7 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The solution of initial value problem.

Solution Summary: The author explains how to solve the initial value problem by applying the laplace transformation on both sides.

Videos

To find: The solution of initial value problem.

Want to see the full answer?

Chapter 7 Solutions

To find:

The solution of initial value problem.