Chapter 17.R, Problem 19E

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Chapter
Section

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

# A Series circuit contains a resistor with R = 40   Ω , an inductor with L = 2  H , a capacitor with C = 0.0025  F , and a 12-V battery. The initial charge is Q = 0.01  C and the initial current is 0. Find the charge at time t.

To determine

To find:

The charge at time t

Explanation

1) Concept:

i. The general second order differential equation for L-C-R circuit is

Ld2Qdt2+RdQdt+1CQ=Et

ii. The general solution of nonhomogeneous differential equation is ay''+by'+cy=Gx can be written as yx=yc+yP

where yc is the general solution of complementary equation; ay''+by'+cy=0 and yP is a particular solution of ay''+by'+cy=Gx

Find yc&yP

2) General solution of ay''+by'+cy=0:

 Roots of ar2+br+c=0 General solution r1,Â r2 are real and distinct y=c1er1x+c2er2x r1=r2=r y=c1erx+c2xerx r1Â ,Â r2Â complexÂ :Î±Â±iÎ² y=eÎ±x(c1cosÎ²x+c2sinÎ²x

3) Method of undetermined coefficient:

It is reasonable to guess that there is a particular solution yP that is a polynomial of the same degree as G because if y is polynomial, then ay''+by'+cy is also polynomial. We therefore substitute yPx= a polynomial into (Of the same degree as ) the differential equation and determine the coefficient.

4) Calculations:

We are given the circuit consist of resistor with R=40Î©, an inductor with L=2Î—, a capacitor with C=0.0025F, and a 12-V battery.

Let Q be the charge in circuit at time t then dQdt be the rate of charge flowwith time t that is current at time t

The general second order differential equation for L-C-R circuit is

Ld2Qdt2+RdQdt+1CQ=Et

By using the given information the differential equation for given L-C-R circuit is

2Q''+40Q'+10.0025Q=12

Given that the initial charge is Q(0)=0.01Â C and the initial current Q'(0) is 0.

Therefore, the given initial value problem is

2Q''+40Q'+400Q=12,Â Â Q0=0.01,Â Â Q'0=0

To find the charge means to find the solution of given initial value problem

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Evaluate the expression sin Exercises 116. (2)3

Finite Mathematics and Applied Calculus (MindTap Course List)

#### Express the function in the form f g. 46. G(x)=x1+x3

Single Variable Calculus: Early Transcendentals, Volume I

#### Place the following set of n = 20 scores in a frequency distribution table.

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

#### Multiply: 3709731

Elementary Technical Mathematics