Question

Laplace transform of f(t)=e^(2t) sinh 3t

Answer 1

The laplace transform of ( e^2t sinh3t ) is *

Answer 2

Answer:

The Laplace transform of f(t) = e^(2t) sinh3t is  L{f(t)} = 3/(s-5)(s+1)

Step-by-step explanation:

Laplace transform:

• Laplace transform is an integral transform that converts a real variable's function to a complex variable's function.
• In mathematics Laplace transform is named after the scientist Pierre-Simon Laplace.

• Given f(t) = e^(2t) sinh3t

         we know that sinh3x = (e^(3x)-e^(-3x))/2

       So, f(t) = e^(2t)[(e^(3t)-e^(-3t))/2]

                  = [(e^(2t)e^(3t)) - (e^(2t)e^(-3t))]/2

             f(t) = [e^(5t) - e^(-t)]/2                          [ since eᵃ. eᵇ = e^(a+b)]

       Now apply Laplace transform

          L{f(t)} = L{[e^(5t) - e^(-t)]/2}

                   = (1/2)[ L{e^(5t)} -  L{e^(-t)} ]

                   = (1/2) [(1/(s-5)) - (1/(s+1))]

                   = (1/2) [ (s+1-s+5)/(s-5)(s+1)]

                   = (6/2)((s-5)(s+1))

          L{f(t)} = 3/(s-5)(s+1)

   Hence, the Laplace transform of f(t) = e^(2t) sinh3t is

                         L{f(t)} = 3/(s-5)(s+1)

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