Ordinary Differential Equations/Laplace Transform

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Definition[edit]

Let be a function on . The Laplace transform of is defined by the integral

The domain of is all values of such that the integral exists.

Existence[edit]

Properties[edit]

Linearity[edit]

Let and be functions whose Laplace transforms exist for and let and be constants. Then, for ,

which can be proved using the properties of improper integrals.

Shifting in s[edit]

If the Laplace transform exists for , then

for .

Proof.

Laplace Transform of Higher-Order Derivatives[edit]

If , then

Proof:
(integrating by parts)

Using the above and the linearity of Laplace Transforms, it is easy to prove that

Derivatives of the Laplace Transform[edit]

If , then

Laplace Transform of Few Simple Functions[edit]

Inverse Laplace Transform[edit]

Definition[edit]

Linearity[edit]