Explanation Given: The provided differential equation is, d 2 x d t 2 + d x d t + d y d t = 0 d 2 y d t 2 + d y d t − 4 d x d t = 0 Provided initial values are x ( 0 ) = 1 , x ' ( 0 ) = 0 y ( 0 ) = − 1 , y ' ( 0 ) = 5 Concept: Linearity property of Laplace transformation is as below. L [ a f ( t ) + b g ( t ) ] = a L { f ( t ) } + b L { g ( t ) } ...... (1) Where a and b are any constants and f and g be any functions of t . The formula for the transform of a derivative in Laplace transformation, L { f ( n ) ( t ) } = s n F ( s ) − s n − 1 f ( 0 ) − s n − 2 f ' ( 0 ) ⋯ − f ( n − 1 ) ( 0 ) Where F ( s ) = L { f ( t ) } The formula for the inverse Laplace transform of e − a t as below. L − 1 { 1 s + a } = e − a t Calculation: Use the above definition to write the Laplace transform of the given differential equation. L { d 2 x d t 2 + d x d t + d y d t } = L { 0 } L { d 2 y d t 2 + d y d t − 4 d x d t } = L { 0 } Use the linearity property of Laplace transformation. L { d 2 x d t 2 } + L { d x d t } + L { d y d t } = L { 0 } L { d 2 y d t 2 } + L { d y d t } − 4 L { d x d t } = L { 0 } Use formula for Laplace of derivatives, L { x ' } | t = 0 = s L { x } − f ( 0 ) = s L { x } And L { y ′ } | t = 0 = s L { y } − f ( 0 ) = s L { y } − 1 Substitute the Laplace transformation of the derivatives in linearity property, ( s + 1 ) L { x } + L { y } = 1 4 L { x } − ( s + 1 ) L { y } = 1 Simplify further, and let L { x } = X ( s ) and L { y } = Y ( s ) ( s + 1 ) X { s } + Y { s } = 1 4 X { s } − ( s + 1 ) Y { s } = 1 Solve the above equations by eliminate, X ( s ) = s + 2 s 2 + 2 s + 5 Y ( s ) = − s + 3 s 2 + 2 s + 5 Write X ( s ) as a sum of partial fractions

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 4.6, Problem 8E

To determine

To Solve: The system of differential equation with the use of Laplace transforms.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.6, Problem 7E

Chapter 4.6, Problem 9E

Students have asked these similar questions

W AutoSave Off Soal Latihan Matdis (1) ▼ Search File Home Insert Draw Design Layout References Mailings Review View Help Aptos (Body) ✓ 12 A A Aa Ро Paste BI U ab x, x² A ✓ A ད Clipboard ₪ 24 23 22 21 20 19 18 17 16 15 1″ ידידיו Page 1 of 1 25°C 215 words Berawan E> M Font 四 Paragraph 3 4 1 56 ☑ 781 LI Comments Editing Find ✓ Normal No Spacing Heading Replace Add-ins Select Styles ☑ Editing Add-ins 91 10 111 12 | 13| 14 15 5. Suppose you wanted to draw a quadrilateral using the dots below as vertices (corners). The dots are spaced one unit apart horizontally and two units apart vertically. a) How many quadrilaterals are possible? b) How many are squares? How many are rectangles? c) How many are parallelograms? English (Indonesia) Accessibility: Investigate R - W ☑ Share ▾ Focus + 100% 00:17 13/04/2025

2. Show that 8 xa S -dx (b² + 12) dr = 2 cos(π2) пра-1 a, b real and -1 0 Your solution should clearly explain the closed contour you are using, and state clearly any vanishing properties of integrals over contours that are being used. You are free to quote from the lectures, the appropriate results on such vanishing properties, without deriving these properties. Any residue calculations involved should be explained clearly.

1. Calculate the integral 500 x sin x (a² +x2)20 dx by using the residue theorem. You may assume that a is real and a > 0. Your solution should clearly explain the closed contour you are using, and state clearly any vanishing properties of integrals over contours that are being used.

Chapter 4 Solutions

Advanced Engineering Mathematics

Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Prob. 18E Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.1 - Prob. 46E Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Prob. 50E Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Prob. 53E Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - Prob. 57E Ch. 4.1 - Prob. 58E Ch. 4.2 - Prob. 1E Ch. 4.2 - Prob. 2E Ch. 4.2 - Prob. 3E Ch. 4.2 - Prob. 4E Ch. 4.2 - Prob. 5E Ch. 4.2 - Prob. 6E Ch. 4.2 - Prob. 7E Ch. 4.2 - Prob. 8E Ch. 4.2 - Prob. 9E Ch. 4.2 - Prob. 10E Ch. 4.2 - Prob. 11E Ch. 4.2 - Prob. 12E Ch. 4.2 - Prob. 13E Ch. 4.2 - Prob. 14E Ch. 4.2 - Prob. 15E Ch. 4.2 - Prob. 16E Ch. 4.2 - Prob. 17E Ch. 4.2 - Prob. 18E Ch. 4.2 - Prob. 19E Ch. 4.2 - Prob. 20E Ch. 4.2 - Prob. 21E Ch. 4.2 - Prob. 22E Ch. 4.2 - Prob. 23E Ch. 4.2 - Prob. 24E Ch. 4.2 - Prob. 25E Ch. 4.2 - Prob. 26E Ch. 4.2 - Prob. 27E Ch. 4.2 - Prob. 28E Ch. 4.2 - Prob. 29E Ch. 4.2 - Prob. 30E Ch. 4.2 - Prob. 31E Ch. 4.2 - Prob. 32E Ch. 4.2 - Prob. 33E Ch. 4.2 - Prob. 34E Ch. 4.2 - Prob. 35E Ch. 4.2 - Prob. 36E Ch. 4.2 - Prob. 37E Ch. 4.2 - Prob. 38E Ch. 4.2 - Prob. 39E Ch. 4.2 - Prob. 40E Ch. 4.2 - Prob. 41E Ch. 4.2 - Prob. 42E Ch. 4.2 - Prob. 45E Ch. 4.2 - Prob. 46E Ch. 4.2 - Prob. 47E Ch. 4.2 - Prob. 48E Ch. 4.2 - Prob. 49E Ch. 4.2 - Prob. 50E Ch. 4.2 - Prob. 51E Ch. 4.2 - Prob. 52E Ch. 4.3 - Prob. 1E Ch. 4.3 - Prob. 2E Ch. 4.3 - Prob. 3E Ch. 4.3 - Prob. 4E Ch. 4.3 - Prob. 5E Ch. 4.3 - Prob. 6E Ch. 4.3 - Prob. 7E Ch. 4.3 - Prob. 8E Ch. 4.3 - Prob. 9E Ch. 4.3 - Prob. 10E Ch. 4.3 - Prob. 11E Ch. 4.3 - Prob. 12E Ch. 4.3 - Prob. 13E Ch. 4.3 - Prob. 14E Ch. 4.3 - Prob. 15E Ch. 4.3 - Prob. 16E Ch. 4.3 - Prob. 17E Ch. 4.3 - Prob. 18E Ch. 4.3 - Prob. 19E Ch. 4.3 - Prob. 20E Ch. 4.3 - Prob. 21E Ch. 4.3 - Prob. 22E Ch. 4.3 - Prob. 23E Ch. 4.3 - Prob. 24E Ch. 4.3 - Prob. 25E Ch. 4.3 - Prob. 26E Ch. 4.3 - Prob. 27E Ch. 4.3 - Prob. 28E Ch. 4.3 - Prob. 29E Ch. 4.3 - Prob. 30E Ch. 4.3 - Prob. 31E Ch. 4.3 - Prob. 32E Ch. 4.3 - Prob. 33E Ch. 4.3 - Prob. 34E Ch. 4.3 - Prob. 35E Ch. 4.3 - Prob. 36E Ch. 4.3 - Prob. 37E Ch. 4.3 - Prob. 38E Ch. 4.3 - Prob. 39E Ch. 4.3 - Prob. 40E Ch. 4.3 - Prob. 41E Ch. 4.3 - Prob. 42E Ch. 4.3 - Prob. 43E Ch. 4.3 - Prob. 44E Ch. 4.3 - Prob. 45E Ch. 4.3 - Prob. 46E Ch. 4.3 - Prob. 47E Ch. 4.3 - Prob. 48E Ch. 4.3 - Prob. 49E Ch. 4.3 - Prob. 50E Ch. 4.3 - Prob. 51E Ch. 4.3 - Prob. 52E Ch. 4.3 - Prob. 53E Ch. 4.3 - Prob. 54E Ch. 4.3 - Prob. 55E Ch. 4.3 - Prob. 56E Ch. 4.3 - Prob. 57E Ch. 4.3 - Prob. 58E Ch. 4.3 - Prob. 59E Ch. 4.3 - Prob. 60E Ch. 4.3 - Prob. 61E Ch. 4.3 - Prob. 62E Ch. 4.3 - Prob. 63E Ch. 4.3 - Prob. 64E Ch. 4.3 - Prob. 65E Ch. 4.3 - Prob. 66E Ch. 4.3 - Prob. 67E Ch. 4.3 - Prob. 68E Ch. 4.3 - Prob. 69E Ch. 4.3 - Prob. 70E Ch. 4.3 - Prob. 71E Ch. 4.3 - Prob. 72E Ch. 4.3 - Prob. 73E Ch. 4.3 - Prob. 74E Ch. 4.3 - Prob. 75E Ch. 4.3 - Prob. 76E Ch. 4.3 - Prob. 77E Ch. 4.3 - Prob. 78E Ch. 4.3 - Prob. 79E Ch. 4.3 - Prob. 80E Ch. 4.3 - Prob. 82E Ch. 4.3 - Prob. 83E Ch. 4.4 - Prob. 1E Ch. 4.4 - Prob. 2E Ch. 4.4 - Prob. 3E Ch. 4.4 - Prob. 4E Ch. 4.4 - Prob. 5E Ch. 4.4 - Prob. 6E Ch. 4.4 - Prob. 7E Ch. 4.4 - Prob. 8E Ch. 4.4 - Prob. 9E Ch. 4.4 - Prob. 10E Ch. 4.4 - Prob. 11E Ch. 4.4 - Prob. 12E Ch. 4.4 - Prob. 13E Ch. 4.4 - Prob. 14E Ch. 4.4 - Prob. 15E Ch. 4.4 - Prob. 16E Ch. 4.4 - Prob. 17E Ch. 4.4 - Prob. 18E Ch. 4.4 - Prob. 19E Ch. 4.4 - Prob. 20E Ch. 4.4 - Prob. 21E Ch. 4.4 - Prob. 22E Ch. 4.4 - Prob. 23E Ch. 4.4 - Prob. 24E Ch. 4.4 - Prob. 25E Ch. 4.4 - Prob. 26E Ch. 4.4 - Prob. 27E Ch. 4.4 - Prob. 28E Ch. 4.4 - Prob. 29E Ch. 4.4 - Prob. 30E Ch. 4.4 - Prob. 31E Ch. 4.4 - Prob. 32E Ch. 4.4 - Prob. 33E Ch. 4.4 - Prob. 34E Ch. 4.4 - Prob. 35E Ch. 4.4 - Prob. 36E Ch. 4.4 - Prob. 37E Ch. 4.4 - Prob. 38E Ch. 4.4 - Prob. 40E Ch. 4.4 - Prob. 41E Ch. 4.4 - Prob. 42E Ch. 4.4 - Prob. 43E Ch. 4.4 - Prob. 44E Ch. 4.4 - Prob. 45E Ch. 4.4 - Prob. 46E Ch. 4.4 - Prob. 47E Ch. 4.4 - Prob. 48E Ch. 4.4 - Prob. 49E Ch. 4.4 - Prob. 50E Ch. 4.4 - Prob. 51E Ch. 4.4 - Prob. 52E Ch. 4.4 - Prob. 53E Ch. 4.4 - Prob. 54E Ch. 4.4 - Prob. 55E Ch. 4.4 - Prob. 56E Ch. 4.4 - Prob. 57E Ch. 4.4 - Prob. 58E Ch. 4.4 - Prob. 59E Ch. 4.4 - Prob. 60E Ch. 4.4 - Prob. 61E Ch. 4.4 - Prob. 65E Ch. 4.4 - Prob. 67E Ch. 4.4 - Prob. 69E Ch. 4.4 - Prob. 71E Ch. 4.4 - Prob. 72E Ch. 4.5 - Prob. 1E Ch. 4.5 - Prob. 2E Ch. 4.5 - Prob. 3E Ch. 4.5 - Prob. 4E Ch. 4.5 - Prob. 5E Ch. 4.5 - Prob. 6E Ch. 4.5 - Prob. 7E Ch. 4.5 - Prob. 8E Ch. 4.5 - Prob. 9E Ch. 4.5 - Prob. 10E Ch. 4.5 - Prob. 11E Ch. 4.5 - Prob. 12E Ch. 4.5 - Prob. 13E Ch. 4.5 - Prob. 14E Ch. 4.5 - Prob. 15E Ch. 4.5 - Prob. 16E Ch. 4.5 - Prob. 17E Ch. 4.5 - Prob. 18E Ch. 4.6 - Prob. 1E Ch. 4.6 - Prob. 2E Ch. 4.6 - Prob. 3E Ch. 4.6 - Prob. 4E Ch. 4.6 - Prob. 5E Ch. 4.6 - Prob. 6E Ch. 4.6 - Prob. 7E Ch. 4.6 - Prob. 8E Ch. 4.6 - Prob. 9E Ch. 4.6 - Prob. 10E Ch. 4.6 - Prob. 11E Ch. 4.6 - Prob. 12E Ch. 4.6 - Prob. 13E Ch. 4.6 - Prob. 14E Ch. 4.6 - Prob. 15E Ch. 4.6 - Prob. 16E Ch. 4.6 - Prob. 18E Ch. 4.6 - Prob. 19E Ch. 4.6 - Prob. 20E Ch. 4 - Prob. 1CR Ch. 4 - Prob. 2CR Ch. 4 - Prob. 3CR Ch. 4 - Prob. 4CR Ch. 4 - Prob. 5CR Ch. 4 - Prob. 6CR Ch. 4 - Prob. 7CR Ch. 4 - Prob. 8CR Ch. 4 - Prob. 9CR Ch. 4 - Prob. 10CR Ch. 4 - Prob. 11CR Ch. 4 - Prob. 12CR Ch. 4 - Prob. 13CR Ch. 4 - Prob. 14CR Ch. 4 - Prob. 15CR Ch. 4 - Prob. 16CR Ch. 4 - Prob. 17CR Ch. 4 - Prob. 18CR Ch. 4 - Prob. 19CR Ch. 4 - Prob. 20CR Ch. 4 - Prob. 21CR Ch. 4 - Prob. 22CR Ch. 4 - Prob. 23CR Ch. 4 - Prob. 24CR Ch. 4 - Prob. 25CR Ch. 4 - Prob. 26CR Ch. 4 - Prob. 27CR Ch. 4 - Prob. 28CR Ch. 4 - Prob. 29CR Ch. 4 - Prob. 30CR Ch. 4 - Prob. 31CR Ch. 4 - Prob. 32CR Ch. 4 - Prob. 33CR Ch. 4 - Prob. 34CR Ch. 4 - Prob. 35CR Ch. 4 - Prob. 36CR Ch. 4 - Prob. 37CR Ch. 4 - Prob. 38CR Ch. 4 - Prob. 39CR Ch. 4 - Prob. 40CR Ch. 4 - Prob. 41CR Ch. 4 - Prob. 42CR Ch. 4 - Prob. 43CR Ch. 4 - Prob. 44CR Ch. 4 - Prob. 45CR Ch. 4 - Prob. 48CR

Knowledge Booster

$Advanced Math$

Learn more about

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

The system of differential equation with the use of Laplace transforms.

Solution Summary: The author explains the Laplace transformation of differential equation, where aandb are any constants and d

Concept explainers

Videos

To Solve: The system of differential equation with the use of Laplace transforms.

Want to see the full answer?

Chapter 4 Solutions