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Ordinary Differential Equations/Laplace Transform

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Definition

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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

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Properties

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Linearity

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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

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If the Laplace transform exists for , then

for .

Proof.

Laplace Transform of Higher-Order Derivatives

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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

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If , then

Laplace Transform of Few Simple Functions

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Inverse Laplace Transform

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Definition

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Linearity

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