Explanation Given data: The function, f ( t ) = ( cos ( 2 t ) + e − 4 t ) u ( t ) Formula used: Write the general expression to find the Laplace transform of f ( t ) . ℒ [ f ( t ) ] = F ( s ) = ∫ 0 − ∞ f ( t ) e − s t d t (1) Here, s is a complex variable. Write the general expression to calculate for the unit step function. u ( t ) = { 0 , t < 0 1 , t > 0 (2) Calculation: Substitute ( cos ( 2 t ) + e − 4 t ) u ( t ) for f ( t ) in equation (1). ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = ∫ 0 − ∞ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] e − s t d t ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = ∫ 0 − ∞ [ cos ( 2 t ) u ( t ) ] e − s t d t + ∫ 0 − ∞ [ e − 4 t u ( t ) ] e − s t d t (3) For equation (3), apply the unit step function in equation (2). ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = ∫ 0 ∞ [ cos ( 2 t ) ] e − s t d t + ∫ 0 ∞ [ e − 4 t ] e − s t d t = ∫ 0 ∞ [ 1 2 ( e j 2 t + e − j 2 t ) ] e − s t d t + ∫ 0 ∞ e − 4 t − s t d t { ∵ cos θ = 1 2 ( e j θ + e − j θ ) } = 1 2 ∫ 0 ∞ ( e j 2 t e − s t + e − j 2 t e − s t ) d t + ∫ 0 ∞ e − ( s + 4 ) t d t ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = 1 2 ∫ 0 ∞ ( e j 2 t − s t + e − j 2 t − s t ) d t + ∫ 0 ∞ e − ( s + 4 ) t d t (4) Reduce the equation (4) as follows, ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = 1 2 ∫ 0 ∞ ( e − ( s − j 2 ) t + e − ( s + j 2 ) t ) d t + ∫ 0 ∞ e − ( s + 4 ) t d t = 1 2 ∫ 0 ∞ e − ( s − j 2 ) t d t + 1 2 ∫ 0 ∞ e − ( s + j 2 ) t d t + ∫ 0 ∞ e − ( s + 4 ) t d t ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = 1 2 [ e − ( s − j 2 ) t − ( s − j 2 ) ] 0 ∞ + 1 2 [ e − ( s + j 2 ) t − ( s + j 2 ) ] 0 ∞ + [ e − ( s + 4 ) t − ( s + 4 ) ] 0 ∞

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Question

Chapter 15.3, Problem 3PP

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Find the Laplace transform of f(t)=(cos(2t)+e−4t)u(t).

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Chapter 15.2, Problem 2PP

Chapter 15.3, Problem 4PP

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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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