Explanation Given: Voltage, E ( t ) = 10 sin 60 t Capacitance, C = 0.01 F Resistance is, R = 2 Ω Initial charge, Q ( 0 ) = 0 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 voltage drop across the capacitor is Q C ,(where Q is the charge) so, R I + Q C = E ( t ) Calculation: Consider the given equation R I + Q C = E ( t ) Substitute I = d Q d t R d Q d t + Q C = E ( t ) (1) Substitute the given values in equation (1) 2 d Q d t + 1 0.01 Q = 10 sin 60 t 2 d Q d t + 1 0.01 ⋅ 100 100 Q = 10 sin 60 t 2 d Q d t + 100 Q = 10 sin 60 t Divide both sides by 2 d Q d t + 50 Q = 5 sin 60 t (2) Compare the above differential equation to the first order linear differential equation. P ( x ) = 50 Integrating factor is I ( x ) = e ∫ P ( x ) d x Substitute P ( x ) = 50 in integrating factor I ( t ) = e ∫ 50 d t = e 50 ∫ d t = e 50 t Multiply both sides by I ( x ) in differential equation (2) e 50 t d Q d t + e 50 t 50 Q = e 50 t 5 sin 60 t e 50 t d Q d t + d ( e 50 t ) d t Q = 5 e 50 t sin 60 t Apply the product rule in above equation ( e 50 t Q ) ′ = 5 e 50 t sin 60 t Integrate both sides with respect to t

Bundle: Single Variable Calculus: Early Transcendentals, Loose-leaf Version, 8th + Webassign Printed Access Card For Calculus, Multi-term Courses, Life Of Edition

18th Edition

ISBN: 9780357008034

Author: Stewart

Publisher: CENGAGE L

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Chapter 9.5, Problem 30E

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To find: The charge and current at time t

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Bundle: Single Variable Calculus: Early Transcendentals, Loose-leaf Version, 8th + Webassign Printed Access Card For Calculus, Multi-term Courses, Life Of Edition

Ch. 9.1 - Show that y=23ex+e2x is a solution of the...Ch. 9.1 - Prob. 2E Ch. 9.1 - (a) For what values of r does the function y = erx...Ch. 9.1 - (a) For what values of k does the function y = cos...Ch. 9.1 - Which of the following functions are solutions of...Ch. 9.1 - (a) Show that every member of the family of...Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E

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The charge and current at time t

Solution Summary: The author explains how to find the charge and current at time t. The first order linear differential equation is of the form dy,dx,+P(x)y=Q

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