# Properties of the Laplace Transform

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

• Section 7.3 – Properties of the Laplace Transform

## Expected Educational Results

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

## Properties of the Laplace Transform

### Translation in $s$

#### Theorem: Translation in $s$

If the Laplace transform $F(s)=\mathcal{L}\left\{f(t)\right\}(s)$ exists for $s>\alpha$, then $\displaystyle \mathcal{L}\left\{e^{at}f(t)\right\}(s)=F(s-a)$ for $s>\alpha +a$.

#### Investigation 01

Prove the above theorem.

#### Investigation 02

Use the above theorem and the above list of common Laplace transforms to find the Laplace transform following functions:

1. $\displaystyle f(t)=t^2e^{3t}$.
2. $\displaystyle f(t)=e^{-t}\sin{(3t)}$.
3. $\displaystyle f(t)=e^{3t}\cos{(-2t)}$.

### Hyperbolic Functions

#### Definitions

• $\displaystyle \sinh{(at)}=\frac{e^{at}-e^{-at}}{2}$
• $\displaystyle \cosh{(at)}=\frac{e^{at}+e^{-at}}{2}$

#### Investigation 03

Use the linearity property of the Laplace transform to find the following. Simplify the result into a single term.

1. $\displaystyle \mathcal{L}\left\{\sinh{(3t)}\right\}(s)$.
2. $\displaystyle \mathcal{L}\left\{\cosh{(2t)}\right\}(s)$.
3. $\displaystyle \mathcal{L}\left\{e^t\cosh{(2t)}\right\}(s)$.
4. $\displaystyle \mathcal{L}\left\{e^{-2t}\sinh{(3t)}\right\}(s)$.