Explanation 1) Concept: i. The general second order differential equation for L - C - R circuit is L d 2 Q d t 2 + R d Q d t + 1 C Q = E t ii. The general solution of nonhomogeneous differential equation is a y ' ' + b y ' + c y = G x can be written as y x = y c + y P where y c is the general solution of complementary equation; a y ' ' + b y ' + c y = 0 and y P is a particular solution of a y ' ' + b y ' + c y = G x Find y c & y P 2) General solution of a y ' ' + b y ' + c y = 0 : Roots of a r 2 + b r + c = 0 General solution r 1 , r 2 are real and distinct y = c 1 e r 1 x + c 2 e r 2 x r 1 = r 2 = r y = c 1 e r x + c 2 x e r x r 1 , r 2 c o m p l e x : α ± i β y = e α x ( c 1 c o s β x + c 2 s i n β x 3) Method of undetermined coefficient: It is reasonable to guess that there is a particular solution y P that is a polynomial of the same degree as G because if y is polynomial, then a y ' ' + b y ' + c y is also polynomial. We therefore substitute y P x = a polynomial into (Of the same degree as ) the differential equation and determine the coefficient. 4) Given: A series circuit consist of a resister with R = 20 Ω , an inductor with L = 1 H , a capacitor with C = 0.002 F , and a 12 - V battery. The initial charge and current are both zero. 5) Calculation: Substitute the values R = 20 Ω , L = 1 H , C = 0.002 F , and E t = 12 V in L d 2 Q d t 2 + R d Q d t + 1 C Q = E t Therefore, it gives d 2 Q d t 2 + 20 · d Q d t + 1 0.002 Q = 12 s i n 10 t By simplifying, Q ' ' + 20 Q + 500 = 12 Therefore, the given initial value problem is Q ' ' + 20 Q ' + 500 Q = 12 , Q 0 = 0 , Q ' 0 = 0 To find the charge means to find the solution of the given initial value problem. The auxiliary equation of the differential equation Q ' ' + 20 Q ' + 500 Q = 0 is r 2 + 20 r + 500 = 0 Solving the auxiliary equation r 2 + 20 r + 500 = 0 That is, r 2 + 20 r + 200 = 0 r = - 20 ± 400 - 2000 2 = - 20 ± - 1600 2 = - 20 ± 40 i 2 That is, r = - 10 ± 20 i By the given concept, the general solution of Q ' ' + 20 Q ' + 500 Q = 0 is Q t = e - 10 t c 1 cos ⁡ 20 t + c 2 sin ⁡ 20 t Now, to find the particular solution of the given differential equation, By using the method of undermined coefficient, Let Q P = A c o s 10 t + B s i n 10 t be the particular solution of the given differential equation Therefore, it satisfies the given differential equation that is Q P ' ' + 20 Q P ' + 500 Q P = 12 s i n 10 t … … … … … … … … 1 Q P = A c o s 10 t + B sin ⁡ 10 t Differentiate with respect to t . Q P ' = - 10 A s i n 10 t + 10 B c o s 10 t Differentiate again with respect to t

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ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

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Chapter 17.3, Problem 15E

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The charge at time t.

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Calculus (MindTap Course List)

Ch. 17.1 - Solve the differential equation. yy6y=0 Ch. 17.1 - Solve the differential equation. y6y+9y=0 Ch. 17.1 - Solve the differential equation. y+2y=0 Ch. 17.1 - Solve the differential equation. y+y12y=0 Ch. 17.1 - Solve the differential equation. 4y+4y+y=0 Ch. 17.1 - Solve the differential equation. 9y+4y=0 Ch. 17.1 - Solve the differential equation. 3y=4y Ch. 17.1 - Solve the differential equation. y=y Ch. 17.1 - Solve the differential equation. y4y+13y=0 Ch. 17.1 - Prob. 10E

Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Prob. 23E Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Prob. 26E Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Let L be a nonzero real number. a Show that the...Ch. 17.1 - Prob. 34E Ch. 17.1 - Consider the boundary-value problem...Ch. 17.2 - Solve the differential equation or initial-value...Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Prob. 12E Ch. 17.2 - Write a trial solution for the method of...Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - Prob. 17E Ch. 17.2 - Write a trial solution for the method of...Ch. 17.2 - Prob. 19E Ch. 17.2 - Prob. 20E Ch. 17.2 - Prob. 21E Ch. 17.2 - Prob. 22E Ch. 17.2 - Prob. 23E Ch. 17.2 - Prob. 24E Ch. 17.2 - Prob. 25E Ch. 17.2 - Prob. 26E Ch. 17.2 - Prob. 27E Ch. 17.2 - Prob. 28E Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - Prob. 3E Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - For the spring in Exercise 4, find the damping...Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - A series circuit consists of a resistor with R=20,...Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - The figure shows a pendulum with length L and the...Ch. 17.4 - Prob. 1E Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Prob. 3E Ch. 17.4 - Prob. 4E Ch. 17.4 - Prob. 5E Ch. 17.4 - Prob. 6E Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Use power series to solve the differential...Ch. 17.4 - Prob. 9E Ch. 17.4 - Prob. 10E Ch. 17.4 - Prob. 11E Ch. 17.4 - The solution of the initial-value problem...Ch. 17.R - Prob. 1CC Ch. 17.R - Prob. 2CC Ch. 17.R - Prob. 3CC Ch. 17.R - Prob. 4CC Ch. 17.R - Prob. 5CC Ch. 17.R - Prob. 1TFQ Ch. 17.R - Prob. 2TFQ Ch. 17.R - Prob. 3TFQ Ch. 17.R - Prob. 4TFQ Ch. 17.R - Prob. 1E Ch. 17.R - Prob. 2E Ch. 17.R - Prob. 3E Ch. 17.R - Prob. 4E Ch. 17.R - Prob. 5E Ch. 17.R - Prob. 6E Ch. 17.R - Prob. 7E Ch. 17.R - Prob. 8E Ch. 17.R - Prob. 9E Ch. 17.R - Solve the differential equation....Ch. 17.R - Prob. 11E Ch. 17.R - Solve the initial-value problem....Ch. 17.R - Prob. 13E Ch. 17.R - Solve the initial-value problem....Ch. 17.R - Prob. 15E Ch. 17.R - Prob. 16E Ch. 17.R - Prob. 17E Ch. 17.R - Use power series to solve the initial-value...Ch. 17.R - Prob. 19E Ch. 17.R - Prob. 20E Ch. 17.R - Assume that the earth is a solid sphere of uniform...

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The charge at time t .

Solution Summary: The author explains that the charge at time t is Q(t)=e-10t

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The charge at time t.

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Chapter 17 Solutions

The charge at time t .

Solution Summary: The author explains that the charge at time t is Q(t)=e-10t

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To find: The charge at time t.

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Chapter 17 Solutions

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The charge at time t.