To determine The value of L ( f ) using the equation (1) and (2) if the value of f ( t ) = sin 2 ω t . Explanation Formula used: 1. The Laplace transform of the function sin a t is a s 2 + a 2 Theorem used: Laplace transform of derivatives: “The transforms of the first and second derivatives of f ( t ) satisfies the following conditions. L ( f ′ ) = s L ( f ) − f ( 0 ) ⋯ ( 1 ) L ( f ″ ) = s 2 L ( f ) − s f ( 0 ) − f ′ ( 0 ) ⋯ ( 2 ) Where (1) holds if f ( t ) is continuous for all t ≥ 0 and f ′ ( t ) is piecewise continuous on every finite interval on the semi axis t ≥ 0 . Similarly, (2) holds if f and f ′ are continuous for all t ≥ 0 and satisfy the growth restriction and f ″ is piecewise continuous on every finite interval on the semi axis t ≥ 0 .” Calculation: The given equation is f ( t ) = sin 2 ω t . Obtain the first derivative of f ( t ) as shown below. f ( t ) = sin 2 ω t f ′ ( t ) = 2 cos ω t sin ω t f ′ ( t ) = sin 2 ω t Now, take Laplace transforms on both sides of the above equation of the first derivative and apply the above mentioned formula as shown below. L ( f ′ ( t ) ) = L ( sin 2 ω t ) = 2 ω 2 s 2 + 4 ω 2 ⋯ ( 3 ) Substitute the value of L ( f ′ ) from equation (1) in equation (3) as shown below

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ISBN: 9781119664697

Author: Kreyszig

Publisher: WILEY

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Chapter 6.2, Problem 19P

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The value of L(f) using the equation (1) and (2) if the value of f(t)=sin2ωt.

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Chapter 6.2, Problem 18P

Chapter 6.2, Problem 20P

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