Explanation Given data: The differential equation is, d 2 v ( t ) d t 2 + s d v ( t ) d t + 10 v ( t ) = 3 cos 2 t (1) The initial conditions are v ( 0 ) = 1 and d v ( 0 ) d t = v ' ( 0 ) = − 2 . Formula used: Write the general expression for the Laplace transform. F ( s ) = L [ f ( t ) ] (2) Write the general expression for the inverse Laplace transform. f ( t ) = L − 1 [ F ( s ) ] (3) Write the general expressions to find the Laplace transform function. L [ d f d t ] = s F ( s ) − f ( 0 − ) (4) L [ d 2 f d t 2 ] = s 2 F ( s ) − s f ( 0 − ) − f ' ( 0 − ) (5) L [ cos ω t ] = s s 2 + ω 2 (6) Here, s is a complex variable, and ω is the angular frequency. Write the general expressions to find the inverse Laplace transform function. L − 1 [ s s 2 + ω 2 ] = cos ω t (7) L − 1 [ ω s 2 + ω 2 ] = sin ω t (8) L − 1 [ s + a ( s + a ) 2 + ω 2 ] = e − a t cos ω t (9) Here, s + a is the frequency shift or frequency translation. Calculation: Apply Laplace transform function given in equation (2), (4), (5) and (6) to equation (1). [ s 2 V ( s ) − s v ( 0 − ) − v ' ( 0 − ) ] + 2 [ s V ( s ) − v ( 0 − ) ] + 10 V ( s ) = 3 ( s s 2 + 2 2 ) ( [ s 2 V ( s ) − s v ( 0 ) − v ' ( 0 ) ] + 2 [ s V ( s ) − v ( 0 ) ] + 10 V ( s ) ) = 3 s s 2 + 2 2 { ∵ v ( 0 − ) = v ( 0 ) , v ' ( 0 − ) = v ' ( 0 ) } (10) Substitute 1 for v ( 0 ) and − 2 for v ' ( 0 ) in equation (10). [ s 2 V ( s ) − s ( 1 ) − ( − 2 ) ] + 2 [ s V ( s ) − 1 ] + 10 V ( s ) = 3 s s 2 + 4 s 2 V ( s ) + 2 s V ( s ) + 10 V ( s ) = 3 s s 2 + 4 − 2 + 2 + s ( s 2 + 2 s + 10 ) V ( s ) = 3 s s 2 + 4 + s ( s 2 + 2 s + 10 ) V ( s ) = 3 s + s ( s 2 + 4 ) s 2 + 4 (11) Rearrange the equation (11) to find V ( s ) . V ( s ) = 3 s + s ( s 2 + 4 ) ( s 2 + 4 ) ( s 2 + 2 s + 10 ) = 3 s + s 3 + 4 s ( s 2 + 4 ) ( s 2 + 2 s + 10 ) V ( s ) = s 3 + 7 s ( s 2 + 4 ) ( s 2 + 2 s + 10 ) (12) Expand V ( s ) using partial fraction. V ( s ) = s 3 + 7 s ( s 2 + 4 ) ( s 2 + 2 s + 10 ) = A s + B s 2 + 4 + C s + D s 2 + 2 s + 10 (13) Here, A , B , C , and D are the constants. Now, to find the constants by using algebraic method. Consider the partial fraction, s 3 + 7 s ( s 2 + 4 ) ( s 2 + 2 s + 10 ) = A s + B s 2 + 4 + C s + D s 2 + 2 s + 10 s 3 + 7 s ( s 2 + 4 ) ( s 2 + 2 s + 10 ) = ( A s + B ) ( s 2 + 2 s + 10 ) + ( C s + D ) ( s 2 + 4 ) ( s 2 + 4 ) ( s 2 + 2 s + 10 ) s 3 + 7 s = ( A s + B ) ( s 2 + 2 s + 10 ) + ( C s + D ) ( s 2 + 4 ) s 3 + 7 s = A s 3 + 2 A s 2 + 10 A s + B s 2 + 2 B s + 10 B + C s 3 + 4 C s + D s 2 + 4 D Reduce the equation as follows, s 3 + 7 s = ( A + C ) s 3 + ( 2 A + B + D ) s 2 + ( 10 A + 2 B + 4 C ) s + ( 10 B + 4 D ) (14) Equating the coefficients of s 3 in equation (14). 1 = A + C C = 1 − A (15) Equating the coefficients of s 2 in equation (14). 0 = 2 A + B + D (16) Equating the coefficients of s in equation (14). 7 = 10 A + 2 B + 4 C (17) Equating the coefficients of constant term in equation (14). 0 = 10 B + 4 D 4 D = − 10 B D = − 10 4 B (18) Substitute equation (18) in equation (16). 0 = 2 A + B − 10 4 B 0 = 2 A + ( 1 − 10 4 ) B 0 = 2 A + ( 4 − 10 4 ) B 0 = 2 A − 6 4 B Reduce the equation as follows, 2 A = 6 4 B A = 6 ( 2 × 4 ) B A = 6 8 B (19) Substitute equation (15) in equation (17). 7 = 10 A + 2 B + 4 ( 1 − A ) 7 − 4 = 10 A + 2 B − 4 A 3 = 6 A + 2 B (20) Substitute equation (19) in equation (20)

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 15, Problem 50P

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Find the time-varying function v(t) for the given differential equation using Laplace transform method.

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Chapter 15 Solutions

Fundamentals of Electric Circuits

Ch. 15.2 - Prob. 1PP Ch. 15.2 - Prob. 2PP Ch. 15.3 - Prob. 3PP Ch. 15.3 - Prob. 4PP Ch. 15.3 - Prob. 5PP Ch. 15.3 - Prob. 6PP Ch. 15.3 - Obtain the initial and the final values of...Ch. 15.4 - Prob. 8PP Ch. 15.4 - Find f(t) if F(s)=48(s+2)(s+1)(s+3)(s+4)Ch. 15.4 - Obtain g(t) if G(s)=s3+2s+6s(s+1)2(s+3)

Ch. 15.4 - Find g(t) given that G(s)=20(s+1)(s2+4s+13)Ch. 15.5 - Graphically convolve the two functions in Fig....Ch. 15.5 - Given g(t) and f(t) in Fig. 15.20, graphically...Ch. 15.5 - Use convolution to find vo(t) in the circuit of...Ch. 15.6 - Prob. 15PP Ch. 15.6 - Prob. 16PP Ch. 15 - Prob. 1RQ Ch. 15 - Prob. 2RQ Ch. 15 - Prob. 3RQ Ch. 15 - Prob. 4RQ Ch. 15 - Prob. 5RQ Ch. 15 - If F(s) = 1/(s + 2), then f(t) is (a) e2t u(t) (b)...Ch. 15 - Prob. 7RQ Ch. 15 - Prob. 8RQ Ch. 15 - Prob. 9RQ Ch. 15 - Prob. 10RQ Ch. 15 - Prob. 1P Ch. 15 - Prob. 2P Ch. 15 - Prob. 3P Ch. 15 - Prob. 4P Ch. 15 - Prob. 5P Ch. 15 - Prob. 6P Ch. 15 - Prob. 7P Ch. 15 - Prob. 8P Ch. 15 - Prob. 9P Ch. 15 - Prob. 10P Ch. 15 - Find F(s) if: (a) ft=6etcosh2t (b) ft=3te2tsinh4t...Ch. 15 - If g(t) = 4e 2t cos 4t, find G(s).Ch. 15 - Prob. 13P Ch. 15 - Prob. 14P Ch. 15 - Prob. 15P Ch. 15 - Prob. 16P Ch. 15 - Prob. 17P Ch. 15 - Prob. 18P Ch. 15 - Prob. 19P Ch. 15 - Prob. 20P Ch. 15 - Prob. 21P Ch. 15 - Prob. 22P Ch. 15 - Prob. 23P Ch. 15 - Design a problem to help other students better...Ch. 15 - Let F(s)=18(s+1)(s+2)(s+3) (a) Use the initial and...Ch. 15 - Determine the initial and final values of f(t), if...Ch. 15 - Prob. 27P Ch. 15 - Prob. 28P Ch. 15 - Prob. 29P Ch. 15 - Prob. 30P Ch. 15 - Find f(t) for each F(s): (a) 10ss+1s+2s+3 (b)...Ch. 15 - Prob. 32P Ch. 15 - Prob. 33P Ch. 15 - Prob. 34P Ch. 15 - Obtain f(t) for the following transforms: (a)...Ch. 15 - Prob. 36P Ch. 15 - Prob. 37P Ch. 15 - Prob. 38P Ch. 15 - Determine f(t) if: (a)...Ch. 15 - Show that...Ch. 15 - Prob. 41P Ch. 15 - Design a problem to help other students better...Ch. 15 - Prob. 43P Ch. 15 - Prob. 44P Ch. 15 - Given h(t) = 4e2tu(t) and x(t) = (t) 2e 2tu(t),...Ch. 15 - Given the following functions...Ch. 15 - A system has the transfer function...Ch. 15 - Find f(t) using convolution given that: (a)...Ch. 15 - Prob. 49P Ch. 15 - Prob. 50P Ch. 15 - Given that v(0) = 5 and dv(0)/dt = 10, solve...Ch. 15 - Prob. 52P Ch. 15 - Prob. 53P Ch. 15 - Design a problem to help other students better...Ch. 15 - Prob. 55P Ch. 15 - Solve for v(t) in the integrodifferential equation...Ch. 15 - Prob. 57P Ch. 15 - Given that dvdt+2v+50tv()d=4u(t) with v(0) = 1,...Ch. 15 - Solve the integrodifferential equation...Ch. 15 - Prob. 60P

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Find the time-varying function v ( t ) for the given differential equation using Laplace transform method.

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Find the time-varying function v(t) for the given differential equation using Laplace transform method.

Chapter 15 Solutions