Laplace Transform Tables & Properties

This worksheet may help you to know about the Laplace Transformation. By using this Laplace transformation we can find particular solution directly without finding the values of arbitrary constants. Generally transform means a device which converts one function into another function. Let f(t) be a continuous function of t defined for all t>=0. Then the Laplace Transform of f(t) is defined as

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L(f(t)) = ∫ e(-st)f(t)dt = F(s)

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Some of the Standard Results and Properties to be remembered:
L(tn) = n! / Sn + 1
L(eat) = 1 / s - a
L(e-at) = 1 / s + a
L(sin at) = a / s² + a²
L(cos at) = s / s² + a²
L(sin hat) = a / s² - a²
L(cos hat) = s / s² + a²
L(1) = 1 / s

Laplace Transform Example
Find L(e^5t (t + 2)²)

Solution:
L(e^5t (t + 2)²) = Lts _{-> s - 5L} ((t + 2)²)
= Lts_{->s - 5} L(t²+4+4t)
= Lts_{->s - 5} [L(t²)+4L(1)+4L(t))]
= Lts_{->s - 5} [2! / s³ + 4 / s + 4 / s²]
= [2 / (s - 5)³] + [4/ s - 5] + [4 / (s - 5)²]

The above worksheet is a walk through to understand the concept of Laplace Transform, Properties and table.