# Laplace Transforms

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

• Section 7.2 – Definition of the Laplace Transform

## Expected Educational Results

• 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.

## Laplace Transforms

Definition: Laplace Transforms

Let $f(t)$ be a function on $[0,\infty)$. The Laplace Transform of $f(t)$ is the function $F(s)$ defined as

$\displaystyle F(s)=\mathcal{L}\left\{f(t)\right\}(s)\equiv\int_0^{\infty}{e^{-st}f(t)\,dt}$

### Exponential Order $\alpha$

#### Activity 06

Use the DESMOS graph below:

For each of the following $f(t)$, find $M$, $\alpha$, and $t_0$, if possible, such that $\left|f(t)\right|\leq Me^{\alpha t}$, for all $t\geq t_0$. Explain.

1. $\displaystyle f(t)=\sin{(2t)}$.
2. $\displaystyle f(t)=30t+2$.
3. $\displaystyle f(t)=2^t$.
4. $\displaystyle f(t)=e^{5t}$.
5. $\displaystyle f(t)=e^{t^2}$.

### Properties of Laplace Transforms

#### Defintion: Exponential Order, $\alpha$

A $f(t)$ is said to be of exponential order $\alpha$ if there exists constants $t_0>0$ and $M>0$ such that for all $t\geq t_0$ $\displaystyle \left|f(t)\right|\leq Me^{\alpha t}$

#### Definition: Piecewise Continuity

A $f(t)$ is said to be piecewise continuous on a finite interval $[a,b]$ if $f(t)$ is continuous at every point in $[a,b]$, except possibly for a finite number of points at which $f(t)$ has a jump discontinuity. A $f(t)$ is said to be piecewise continuous on $[0,\infty)$ if $f(t)$ is piecewise continuous on $[0,N]$ for all $N>0$.

#### Theorem: Existence of Laplace Transform

If $f(t)$ is piecewise continuous on $[0,\infty)$ and of exponential order $\alpha$, then $\mathcal{L}\left\{f(t)\right\}(s)$ exists for $s>\alpha$.

#### Theorem: Linearity of the Laplace Transform

Let $f(t)$, $f_1(t)$, and $f_2(t)$ be functions with exponential order alpha. Let $c$ be a constant. Then for $s>\alpha$ $\mathcal{L}\left\{f_1(t)+f_2(t)\right\}(s)=\mathcal{L}\left\{f_1(t)\right\}(s)+\mathcal{L}\left\{f_2(t)\right\}(s)$ $\mathcal{L}\left\{cf(t)\right\}(s)=c\mathcal{L}\left\{f(t)\right\}(s)$.

#### Investigation 07

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

#### Investigation 08

Find the Laplace transform following functions:

1. $\displaystyle f(t)=3e^{2t}-t^2$.
2. $\displaystyle f(t)=5t^2e^{3t}+\sin{(4t)}$.
3. $\displaystyle f(t)=e^{3t}\sin{(2t)}+5-2te^{t}$.
4. $\displaystyle f(t)=e^{t}-\cos{(3t)}+t^2$.
5. $\displaystyle f(t)=4te^{t}-7e^{2t}\cos{(5t)}+7$.

#### Use Technology to Verify Laplace Transform

Mathematica

1(* Example from Investigation 09-1 *)2(* Find Laplace transform of f(t)=3e^(2t)-t^2 *)3LaplaceTransform[3*E^(2t)-t^2, t, s]

Warnings:

1. 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.

2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

3. The arguments for LaplaceTransform[ ] are:

1. the function that the transform is being applied;
2. the independent variable of the function;
3. the independent variable for the transform.