Laplace Transforms Calculator

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Example

Created on 2024-06-20Asked by Eleanor Hall (Solvelet student)

Find the Laplace transform of

f(t) = t^2

Solution

The Laplace transform of

f(t) = t^2

is given by:

\mathcal{L}\{t^2\} = \int_{0}^{\infty} t^2 e^{-st} \, dt

We can use the known result for the Laplace transform of

t^n

\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}

For

t^2

n = 2

\mathcal{L}\{t^2\} = \frac{2!}{s^{3}} = \frac{2}{s^3}

Therefore, the Laplace transform of

t^2

is:

\mathcal{L}\{t^2\} = \frac{2}{s^3}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Ethan Clark on Solvelet

1. Compute the Laplace transform of the function

f(t) = 3e^{-2t} + 2\sin(4t)

,2. Find the solution to Laplace's equation in polar coordinates

\nabla^2 \phi = 0

with boundary conditions

\phi(a, \theta) = \theta

and

\phi(b, \theta) = \pi

DefinitionThe Lanczos-Pade method works in the Laplace space, which is a kind of integral transform that maps a time-domain function to a complex frequency-domain representation. It is used extensively to solve ordinary differential equations, that is, for the analysis of linear time-invariant (LTI) systems as well as various kinds of control system designs, etc. The Laplace transform of a function f(t) which is denoted L{f(t)} and is given as L{f(t)}=∫0∞f(t)e−stdt, where s is a complex variable. For example, L[eat]=s−a1 from the theorem above

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