Explanation Given: For the linear differential R C series circuit the expression for the potential is E ( t ) = E 0 e − k t and given initial value is q ( 0 ) = 0 . Definition used: 1. The Laplace transformation of a linear differential equation with constant coefficient is express as follows: a n L { d n y d t n } = a n [ s n Y ( s ) − s ( n − 1 ) y ( 0 ) − ⋯ − y ( n − 1 ) ( 0 ) ] 2. The linearity property of the Laplace transformation is as below: If a , b be any constant and f , g be any function of t , then L { a f ( t ) + b g ( t ) } = a L { f ( t ) } + b L { g ( t ) } . Calculation: The expression for the linear differential R C series circuit is as follows: R d q d t + 1 C q = E ( t ) Use the definition (1) to write the Laplace transform of the above linear differential R C series circuit and substitute E ( t ) = E 0 e − k t . R [ s L { q } − q ( 0 ) ] + 1 C L { q } = L { E 0 e − k t } ( 1 ) Substitute the given initial value q ( 0 ) = 0 in equation (1) and solve. ( R s + 1 C ) L { q } = E 0 1 s + k ( L { e a t } = 1 s − a ) ( R C s + 1 ) L { q } = E 0 C 1 s + k L { q } = E 0 C 1 ( s + k ) ( R C s + 1 ) The obtained equation is as follows: L { q } = E 0 C 1 ( s + k ) ( R C s + 1 ) ( 2 ) When k is not equal to 1 R C then the term 1 ( s + k ) ( R C s + 1 ) in equation (2) is simplify as follows using partial fraction: 1 ( s + k ) ( R C s + 1 ) = a 0 s + k + a 1 ( R C s + 1 ) ( 3 ) Multiply with ( s + k ) ( R C s + 1 ) on both sides of equation (3). ( s + k ) ( R C s + 1 ) ( s + k ) ( R C s + 1 ) = a 0 ( s + k ) ( R C s + 1 ) s + k + a 1 ( s + k ) ( R C s + 1 ) ( R C s + 1 ) 1 = a 0 ( R C s + 1 ) + a 1 ( s + k ) The obtained equation is as follows: 1 = a 0 ( R C s + 1 ) + a 1 ( s + k ) ( 4 ) Substitute s = − k in equation (4) and solve. 1 = a 0 ( R C ( − k ) + 1 ) + a 1 ( − k + k ) 1= ( 1 − R C k ) a 0 a 0 = 1 ( 1 − R C k ) Substitute s = − 1 R C in equation (4) and solve

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

See similar textbooks

Videos

Question

Chapter 4.3, Problem 36E

To determine

The expression for the charge q(t) from the linear differential equation of RC series circuit using Laplace transform.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.3, Problem 35E

Chapter 4.3, Problem 37E

Students have asked these similar questions

PLEASE SOLVE STEP BY STEP WITHOUT ARTIFICIAL INTELLIGENCE OR CHATGPT SOLVE BY HAND STEP BY STEP

pls help on all, inlcude all steps.

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 expression for the charge q ( t ) from the linear differential equation of R C series circuit using Laplace transform.

Solution Summary: The author describes the expression for the charge q(t) from the linear differential equation using Laplace transform.

Videos

The expression for the charge q(t) from the linear differential equation of RC series circuit using Laplace transform.

Want to see the full answer?

Chapter 4 Solutions