Question

Calculate the Laplace transform of the function f(t) = e^(at)·sinh(bt).

Answer 1

The Laplace transform of the function f(t) =
`e^((at)·sinh(bt))` is F(s) = (a / (s - b)² + a²) / (s² - b²).

Step-by-step explanation:

Laplace Transform Properties: The Laplace transform of functions involving sinh and e^t can be found using the following properties:

L{sinh(bt)} = b / (s² - b²)

L{
`e^{(at)`} = 1 / (s - a)

Applying the Properties: Using these properties, we can decompose f(t) and transform each term separately:

L{
`e^((at)·sinh(bt))`} = L{
`e^{(at)`} * L{sinh(bt)}

Transforming the Terms:

L{
`e^{(at)`} = 1 / (s - a)

L{sinh(bt)} = b / (s² - b²)

Combining Terms:

Multiplying the transformed terms and simplifying:

L{
`e^((at)·sinh(bt))`} = (1 / (s - a)) * (b / (s² - b²)) = (a / (s - b)² + a²) / (s² - b²)

Therefore, the Laplace transform of f(t) =
`e^((at)·sinh(bt))` is F(s) = (a / (s - b)² + a²) / (s² - b²).