Explanation Given data: The time-varying function is 5 sin ( a t ) cos ( b t ) . Formula used: Write the general expression to find the Laplace transform of f ( t ) . F ( s ) = ℒ [ f ( t ) ] (1) Here, s is a complex variable. Refer to Table 15.2 in the textbook. Write the general expressions to find the Laplace transform function. ℒ [ e − a t ] = 1 s + a (2) ℒ [ e a t ] = 1 s − a (3) Calculation: Consider the function, f ( t ) = 5 sin ( a t ) cos ( b t ) = 5 [ 1 2 j ( e j a t − e − j a t ) ] [ 1 2 ( e j b t + e − j b t ) ] { ∵ sin θ = 1 2 j ( e j θ − e − j θ ) cos θ = 1 2 ( e j θ + e − j θ ) } = 5 ( 2 j ) ( 2 ) ( e j a t − e − j a t ) ( e j b t + e − j b t ) = − 1.25 j ( e j a t e j b t + e j a t e − j b t − e − j a t e j b t − e − j a t e − j b t ) Reduce the equation as follows. f ( t ) = − 1.25 j ( e j a t + j b t + e j a t − j b t − e − j a t + j b t − e − j a t − j b t ) = − 1.25 j ( e j ( a + b ) t + e j ( a − b ) t − e − j ( a − b ) t − e − j ( a + b ) t ) Substitute − 1.25 j ( e j ( a + b ) t + e j ( a − b ) t − e − j ( a − b ) t − e − j ( a + b ) t ) for f ( t ) in equation (1). F ( s ) = ℒ [ − 1.25 j ( e j ( a + b ) t + e j ( a − b ) t − e − j ( a − b ) t − e − j ( a + b ) t ) ] = ( − 1.25 j ) ℒ [ e j ( a + b ) t + e j ( a − b ) t − e − j ( a − b ) t − e − j ( a + b ) t ] F ( s ) = − 1.25 j ( ℒ [ e j ( a + b ) t ] + ℒ [ e j ( a − b ) t ] − ℒ [ e − j ( a − b ) t ] − ℒ [ e − j ( a + b ) t ] ) (4) Apply Laplace transform function given in equation (2) and (3) to equation (4). F ( s ) = − 1

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Chapter 15, Problem 1P

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Find the Laplace transform of 5sin(at)cos(bt).

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Chapter 15 Solutions

EBK 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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