# Ordinary Differential Equations/Laplace Transform

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## Contents

## 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