The Laplace transform of t^3 is equal to 6/s^4. In this article, we will learn how to find the Laplace transform of t cube.
Table of Contents
Laplace Transform of t Cube Formula
The Laplace transform formula of t cube, that is, the formula of L{t3} is given by
L{t3} = 6/s4.
What is the Laplace Transform of t3?
Answer: The Laplace transform of t3 is 6/s4.
Proof:
To find the Laplace transform of t cube by the definition of Laplace transform, let us recall the definition. The Laplace transform of f(t) is defined by
L{f(t)} = $\int_0^\infty$ f(t) e-st dt.
Step 1: Put f(t) = t3.
Therefore,
L{t3} = $\int_0^\infty$ t3 e-st dt. …(I)
Step 2: We 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.
| t | z |
| 0 | 0 |
| ∞ | ∞ |
Step 3: Therefore, from (I) we get that
L{t3} = $\int_0^\infty \Big(\dfrac{z}{s} \Big)^3 e^{-z} \dfrac{dz}{s}$
= (1/s3+1) $\int_0^\infty z^{3+1-1} e^{-z} dz$
= (1/s4) $\Gamma(3+1)$, by the definition of the Gamma function.
= (1/s4) × 3! as we know that Γ(n+1) = n!
= 3!/s4
= 6/s4.
Therefore, the Laplace transform of t^3 is equal to 6/s4 and this is proved by the definition of Laplace transforms.
Question-Answer
| Question: What is the inverse Laplace transform of 1/s4? |
Answer:
From above, the Laplace transform of t3 is equal to 6/s4. That is,
L{t3} = $\dfrac{6}{s^4}$
Taking inverse Laplace on both sides, we obtain that
t3 = $L^{-1}\left\{\dfrac{6}{s^4} \right\}$
⇒ t3 = $6L^{-1}\left\{\dfrac{1}{s^4} \right\}$
⇒ $\dfrac{t^3}{6}= L^{-1}\left\{\dfrac{1}{s^4} \right\}$.
Thus, the inverse Laplace transform of 1/s4 is equal to t3/6.
Read Also:
Concept of Laplace Transform: Definition, Table, Formulas, Properties & Examples
Laplace Transform of eat: The Laplace transform of eat is 1/(s-a).
Laplace transform of constant: The Laplace transform of c is c/s.
Laplace transform of sin(at): The Laplace transform of sin(at) is a/(s2+a2).
Laplace transform of cos(at): The Laplace transform of cos(at) is s/(s2+a2).
Inverse Laplace transform of constant: The inverse Laplace transform of c is cδ(t), where δ(t) is the Dirac delta function.
FAQs
Answer: The Laplace transform of t cube is equal to 6/s4.
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