Explanation Given data: The integrodifferential equation is, d y d t + 3 y ( t ) + 2 ∫ 0 t y ( τ ) d τ = 2 e − 3 t (1) The initial condition y ( 0 ) is 0 . 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 [ ∫ 0 t f ( x ) d x ] = 1 s F ( s ) (5) L [ e − a t u ( t ) ] = 1 s + a (6) Here, s is a complex variable, and s + a is the frequency shift or frequency translation. Write the general expressions to find the inverse Laplace transform function. L − 1 [ 1 s + a ] = e − a t u ( t ) (7) Calculation: Apply Laplace transform function given in equation (2), (4), (5) and (6) to equation (1). [ s Y ( s ) − y ( 0 − ) ] + 3 Y ( s ) + 2 ( 1 s Y ( s ) ) = 2 ( 1 s + 3 ) [ s Y ( s ) − y ( 0 ) ] + 3 Y ( s ) + 2 s Y ( s ) = 2 s + 3 { ∵ y ( 0 − ) = y ( 0 ) } (8) Substitute 0 for y ( 0 ) in equation (8). [ s Y ( s ) − 0 ] + 3 Y ( s ) + 2 s Y ( s ) = 2 s + 3 ( s + 3 + 2 s ) Y ( s ) = 2 s + 3 ( s 2 + 3 s + 2 s ) Y ( s ) = 2 s + 3 (9) Rearrange the equation (9) to find Y ( s ) . Y ( s ) = ( 2 s + 3 ) ( s 2 + 3 s + 2 s ) = 2 s ( s + 3 ) ( s 2 + 3 s + 2 ) = 2 s ( s + 3 ) ( s 2 + 2 s + 1 s + 2 ) = 2 s ( s + 3 ) ( s ( s + 2 ) + 1 ( s + 2 ) ) Reduce the equation as follows, Y ( s ) = 2 s ( s + 3 ) ( s + 2 ) ( s + 1 ) Therefore, Y ( s ) = 2 s ( s + 1 ) ( s + 2 ) ( s + 3 ) (10) Expand Y ( s ) using partial fraction. Y ( s ) = 2 s ( s + 1 ) ( s + 2 ) ( s + 3 ) = A s + 1 + B s + 2 + C s + 3 (11) Here, A , B , and C are the constants. Now, to find the constants by using residue method. Constant A : A = ( s + 1 ) Y ( s ) | s + 1 = 0 (12) Substitute equation (10) in equation (12) to find the constant A

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Chapter 15.6, Problem 16PP

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

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Chapter 15.6, Problem 15PP

Chapter 15, Problem 1RQ

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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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