(a) Explanation Given: Voltage, E ( t ) = 40 sin 60 t volts The inductance, L = 1 H The resistance is R = 20 Ω Initial Current, I ( 0 ) = 1 A Derivative rule: Product rule d ( f ⋅ g ) d x = f ⋅ d ( g ) d x + g ⋅ d ( f ) d x The First order linear differential equation is of the form d y d x + P ( x ) y = Q ( x ) Kirchhoff’s law The sum of voltage drops is equal to the supplied voltage E ( t ) , that is, L d I d t + R I = E ( t ) (1) Calculation: Substitute the given values in equation (1) d I d t + 20 I = 40 sin 60 t Compare the above differential equation to the first order linear differential equation. So the integrating factor is I ( x ) = e ∫ P ( x ) d x Substitute P ( x ) = 20 in integrating factor I ( t ) = e ∫ 20 d t = e 20 ∫ d t = e 20 t Multiply both sides of the differential equation (1) by I ( t ) e 20 t d I d t + 20 e 20 t I = e 20 t 40 sin 60 t Apply the product rule in above equation ( e 20 t I ) ′ = e 20 t 40 sin 60 t Integrate both sides with respect to t . ∫ ( e 20 t I ) ′ d t = ∫ e 20 t 40 sin 60 t d t e 20 t I = 40 ∫ e 20 t sin 60 t d t Use the formula ∫ e a x sin b x d x = 1 a 2 + b 2 e a x ( a sin b x − b cos b x ) + C (b) To determine To find: The current after 0.1 seconds. (c) To determine To sketch: The graph of the current function.

9th Edition

ISBN: 9781337613927

Author: Stewart

Publisher: CENGAGE L

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Chapter 9.5, Problem 32E

(a)

To determine

To find: The value of I (t)

(b)

To determine

To find: The current after 0.1 seconds.

(c)

To determine

To sketch: The graph of the current function.

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Chapter 9.5, Problem 31E

Chapter 9.5, Problem 33E

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Chapter 9 Solutions

Calculus, Early Transcendentals

Ch. 9.1 - Write a differential equation that models the...Ch. 9.1 - Write a differential equation that models the...Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Determine whether the given function is a solution...Ch. 9.1 - Determine whether the given function is a solution...Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 12E Ch. 9.1 - Show that the given function is a solution of the...

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The value of I ( t )

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To find: The value of I (t)

To find: The current after 0.1 seconds.

To sketch: The graph of the current function.

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Chapter 9 Solutions