Laplace Transform: Definition, Properties, and Applications, Lecture notes of Differential Equations

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LAPLACE TRANSFORM

Pierre-Simon Laplace

The Laplace Transform

✓By default, the domain of the function f = f(t) is the set of all nonnegative real numbers. The domain of its Laplace transform depends on f and can vary from a function to a function.

Definition of the Laplace Transform

Let f(t) be defined for t≥ 0; then,

EXAMPLE

F(𝑠) = න

0 ∞ 𝑒 −𝑠𝑡 1 𝑑𝑡 F 𝑠 = −

−𝑠𝑡 F 𝑠 = −

−∞ − −

0 0 ∞ 0 1

EXAMPLE

F( 1 ) = න

0 ∞ 𝑒 −𝑠𝑡 1 𝑑𝑡 F 1 = 0 − −

F 1 =

PROPERTIES OF LAPLACE TRANSFORM

LAPLACE TRANSFORM IS LINEAR

INVERSE

LAPLACE TRANSFORM

INVERSE LAPLACE TRANSFORM

INVERSE LAPLACE TRANSFORM

INVERSE LAPLACE TRANSFORM

EXAMPLE

• Evaluate the inverse of the following: ✓F s = 5 𝑠+ 1 − 6 𝑠 2

• 4 − 1 𝑠 4

TABLE OF TRANSFORMS