Find Laplace transform of t^2 [t Square]

The Laplace transform of t^2, i.e the Laplace of t square is equal to 2/s3. In this article, we will learn how to find the Laplace transform of t2.

The Laplace transform t2 (t square) is denoted by L{t2}, and its formula is given as follows:

L{t2} = 2/s3.

This follows from the Laplace formula of tn: L{tn} = n!/sn+1 by putting n=2.

What is the Laplace Transform of t2?

Answer: The Laplace transform of t2 is equal to 2/s3.

Proof:

The Laplace transform of f(t) by definition is given by

L{f(t)} = $\int_0^\infty$ f(t) e-st dt.

So to find the Laplace transform of t2 by definition, we need to follow below steps.

Step 1: Put f(t) = t2.

Therefore,

L{t2} = $\int_0^\infty$ t2 e-st dt. …(I)

Step 2: In order to compute the above integral, let us use the theory of the Gamma function Γ(x) = $\int_0^\infty$ tx-1 e-t dx. Assume that

z=st

∴ dz=s dt ⇒ dt = dz/s. Also, t=z/s.

Step 3: The equation (I) then implies that

L{t2} = $\int_0^\infty \Big(\dfrac{z}{s} \Big)^2 e^{-z} \dfrac{dz}{s}$

= (1/s2+1) $\int_0^\infty z^{2+1-1} e^{-z} dz$

= (1/s3) $\Gamma(2+1)$, by the definition of the Gamma function.

= (1/s3) × 2! as we know that Γ(n+1) = n!

= 2!/s3

= 2/s3.

Therefore, the Laplace transform of t^2 is equal to 2/s3 and this is proved by the definition of Laplace transforms.