Question

explain steps to solve. Find the Laplace Transform of the function ( x(t) ) given below: [ x(t)=t² e`⁻³ ᵗ} u(t-1) ]

Answer 1

To find the Laplace Transform of the function x(t) = t²e^(-3t)u(t-1), you can use the formula for the Laplace Transform of a function times a unit step function. The Laplace Transform of t^n, where n is a positive integer, is given by n! / s^(n+1).

Step-by-step explanation:

To find the Laplace Transform of the function x(t) = t²e^(-3t)u(t-1), we can use the formula for the Laplace Transform of a function times a unit step function. The Laplace Transform of t^n, where n is a positive integer, is given by n! / s^(n+1). So, the Laplace Transform of t² is 2! / s^3. The Laplace Transform of e^(-at) is 1 / (s + a). And the Laplace Transform of u(t-1) is e^(-s).

Therefore, the Laplace Transform of x(t) = t²e^(-3t)u(t-1) is given by:

X(s) = (2! / s^3) * (1 / (s + 3)) * e^(-s)

So, the Laplace Transform of the given function is X(s) = 2! * e^(-s) / (s^3 * (s + 3)).