Laplace TransformsExpected Educational ResultsLaplace TransformsExponential Order Activity 06Properties of Laplace TransformsDefintion: Exponential Order, Definition: Piecewise ContinuityTheorem: Existence of Laplace TransformTheorem: Linearity of the Laplace TransformInvestigation 07Investigation 08Check Your WorkUse Technology to Verify Laplace TransformCC BY-NC-SA 4.0

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

**Section 7.2**– Definition of the Laplace Transform

**Objective 19–1**: I understand the Laplace transform as a map of a function onto a specific class of functions.**Objective 19–2**: I can use the definition of Laplace transform to find the Laplace Transform of a given any function.

Definition: Laplace Transforms

Let be a function on . The **Laplace Transform** of is the function defined as

Use the DESMOS graph below:

For each of the following , find , , and , if possible, such that , for all . Explain.

- .
- .
- .
- .
- .

A is said to be of **exponential order ** if there exists constants and such that for all

A is said to be **piecewise continuous on a finite interval ** if is continuous at every point in , except possibly for a finite number of points at which has a jump discontinuity. A is said to be **piecewise continuous on ** if is piecewise continuous on for all .

If is piecewise continuous on and of exponential order , then exists for .

Let , , and be functions with exponential order alpha. Let be a constant. Then for .

Assume the conditions are met, prove both statements of the Linearity of the Laplace transform theorem.

Find the Laplace transform following functions:

- .
- .
- .
- .
- .

**Mathematica**

`1``(* Example from Investigation 09-1 *)`

2`(* Find Laplace transform of f(t)=3e^(2t)-t^2 *)`

3`LaplaceTransform[3*E^(2t)-t^2, t, s]`

**Warnings**:

Be very

**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

The arguments for

*LaplaceTransform[ ]*are:- the function that the transform is being applied;
- the independent variable of the function;
- the independent variable for the transform.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Last Modified**: Sunday, 8 November 2020 4:12 EDT