Question

Laplace Transform of Derivatives Given the function f(t) = e ^ (- 2t) * cos(3t) , compute the Laplace transform of its first and second derivatives.

Answer 1

To compute the Laplace transform of the first and second derivatives of the given function, we first find the derivatives of the function and then use the properties of the Laplace transform to compute the Laplace transforms of the derivatives.

Step-by-step explanation:

To compute the Laplace transform of the first and second derivatives of the function f(t) = e^(-2t) * cos(3t), we first need to find the first and second derivatives of f(t).

First derivative of f(t):

f'(t) = -2e^(-2t) * cos(3t) - 3e^(-2t) * sin(3t)

Second derivative of f(t):

f''(t) = 4e^(-2t) * cos(3t) + 9e^(-2t) * sin(3t)

Next, we can use the properties of the Laplace transform to compute the Laplace transform of the derivatives:

L{f'(t)} = -2 * L{e^(-2t) * cos(3t)} - 3 * L{e^(-2t) * sin(3t)}

L{f''(t)} = 4 * L{e^(-2t) * cos(3t)} + 9 * L{e^(-2t) * sin(3t)}

Using the Laplace transform of e^(-at) * cos(bt) and e^(-at) * sin(bt), we can compute the Laplace transforms of the derivatives.