Explanation Given data: The differential equation is, d 2 i d t 2 + 3 d i d t + 2 i + δ ( t ) = 0 (1) The initial conditions are i ( 0 ) = 0 and i ' ( 0 ) = 3 . 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 [ δ ( t ) ] = 1 (6) Here, s is a complex variable. Write the general expressions to find the inverse Laplace transform function. L − 1 [ 1 s + a ] = e − a t u ( t ) (7) 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 I ( s ) − s i ( 0 − ) − i ' ( 0 − ) ] + 3 [ s I ( s ) − i ( 0 − ) ] + 2 I ( s ) + 1 = 0 [ s 2 I ( s ) − s i ( 0 ) − i ' ( 0 ) ] + 3 [ s I ( s ) − i ( 0 ) ] + 2 I ( s ) + 1 = 0 { ∵ i ( 0 − ) = i ( 0 ) , i ' ( 0 − ) = i ' ( 0 ) } (8) Substitute 0 for i ( 0 ) and 3 for i ' ( 0 ) in equation (8). [ s 2 I ( s ) − s ( 0 ) − 3 ] + 3 [ s I ( s ) − 0 ] + 2 I ( s ) + 1 = 0 s 2 I ( s ) − 3 + 3 s I ( s ) + 2 I ( s ) + 1 = 0 s 2 I ( s ) + 3 s I ( s ) + 2 I ( s ) = 3 − 1 ( s 2 + 3 s + 2 ) I ( s ) = 2 (9) Rearrange the equation (9) to find I ( s ) . I ( s ) = 2 s 2 + 3 s + 2 = 2 s 2 + 1 s + 2 s + 2 = 2 ( s + 1 ) s + 2 ( s + 1 ) I ( s ) = 2 ( s + 1 ) ( s + 2 ) (10) Expand I ( s ) using partial fraction

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Chapter 15, Problem 52P

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

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Chapter 15, Problem 51P

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Chapter 15 Solutions

Fundamentals of Electric Circuits

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

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

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