The Laplace transform of t^2 u(t-1) is equal to e^{-s}[2/s^{3} + 2/s^{2} +1/s]. Here we find the Laplace of t^{2}u(t-1) using the second shifting property of Laplace transforms.

The formula of the Laplace of t^{2}u(t-1) is given as

L{t^{2}u(t-1)} = $e^{-s}\Big[ \dfrac{2}{s^3} + \dfrac{2}{s^2} +\dfrac{1}{s}\Big]$.

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## What is the Laplace of t^{2}u(t-1)?

Answer: The Laplace of t^{2}u(t-1) is e^{-s}[2/s^{3} + 2/s^{2} +1/s]. |

**Explanation:**

To find the Laplace of t^{2}u(t-1) we will use the second shifting theorem of Laplace transforms which says that if L{f(t)} = F(s), then

L{f(t-a) u(t-a)} = e^{-as} F(s) for a>0 **…(∗)**

Comparing f(t-a) u(t-a) with the given function t^{2}u(t-1), we get that

a=1 and f(t-1)=t^{2}.

So f(t) = (t+1)^{2} = t^{2}+2t+1.

Hence, F(s) = L{t^{2}+2t+1} = 2/s^{3} + 2/s^{2} +1/s, by the formula L{t^{n}} = n!/s^{n+1}.

Now, using the formula **(∗)**, we obtain that

L{t^{2}u(t-1)} = $e^{-s}\Big[ \dfrac{2}{s^3} + \dfrac{2}{s^2} +\dfrac{1}{s}\Big]$.

So the Laplace transform of t^{2}u(t-1) is equal to e^{-s}[2/s^{3} + 2/s^{2} +1/s], and this is proved by using the second shifting property of Laplace transforms.

Related Topics:

Laplace transform of unit step function, L{u(t)}

Find the Laplace transform of u(t-1)

Find the Laplace transform of u(t-2)

## FAQs

**Q1: What is the Laplace transform of t**

^{2}u(t-1)?Answer: The Laplace transform of t^{2}u(t-1) is equal to e^{-s}[2/s^{3} + 2/s^{2} +1/s], that is, L{t^{2}u(t-1)} = e^{-s}[2/s^{3} + 2/s^{2} +1/s].

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.