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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

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, r2 are real and distinct y=c1er1x+c2er2x
r1=r2=r y=c1erx+c2xerx
r1 , r2 complex :α±iβ y=eαx(c1cosβx+c2sinβx

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=40Ω, an inductor with L=2Η, 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)=0.01 C and the initial current Q'(0) is 0.

Therefore, the given initial value problem is

2Q''+40Q'+400Q=12,  Q0=0.01,  Q'0=0

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

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Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.1 P-35ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ESect-17.R P-1CCSect-17.R P-2CCSect-17.R P-3CCSect-17.R P-4CCSect-17.R P-5CCSect-17.R P-1TFQSect-17.R P-2TFQSect-17.R P-3TFQSect-17.R P-4TFQSect-17.R P-1ESect-17.R P-2ESect-17.R P-3ESect-17.R P-4ESect-17.R P-5ESect-17.R P-6ESect-17.R P-7ESect-17.R P-8ESect-17.R P-9ESect-17.R P-10ESect-17.R P-11ESect-17.R P-12ESect-17.R P-13ESect-17.R P-14ESect-17.R P-15ESect-17.R P-16ESect-17.R P-17ESect-17.R P-18ESect-17.R P-19ESect-17.R P-20ESect-17.R P-21E

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