Properties of the Laplace TransformExpected Educational ResultsProperties of the Laplace TransformLaplace Transform of the DerivativeTheoremInvestigation 04Laplace Transform of the Higher Order DerivativesTheoremDerivatives of the Laplace TransformTheoremInvestigation 05CC BY-NC-SA 4.0

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 7.3**– Properties of the Laplace Transform

**Objective 20–1**: I understand the properties of the Laplace transform.**Objective 20–2**: I can utilize the properties of the Laplace transform.

Let be continuous on and of exponential order . Let be piecewise continuous on and of exponential order . Then for ,

Prove the above theorem.

Let , , , , be continuous on , let be piecewise continuous on , with all functions of exponential order . Then for ,

**NOTE**: The *Method of Mathematical Induction* is required to prove the above theorem.

Essentially, the Laplace transform changes a differential equation in terms of into an algebraic expression in terms of which will be helpful in solving higher-order IVPs.

Let and assume is piecewise continuous on and of exponential order . Then for ,

- Evaluate .
- Evaluate .
- Evaluate .
- Evaluate .

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**Last Modified**: Sunday, 8 November 2020 4:26 EDT