Properties of the Laplace Transform

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:

Expected Educational Results

Properties of the Laplace Transform

Laplace Transform of the Derivative

Theorem

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

Investigation 04

Prove the above theorem.

Laplace Transform of the Higher Order Derivatives

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.

Derivatives of the Laplace Transform

Theorem

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

Investigation 05

  1. Evaluate .
  2. Evaluate .
  3. Evaluate .
  4. Evaluate .

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