# Inverse Laplace Transforms

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

• Section 7.4 – Inverse Laplace Transforms

## Prerequisite Knowledge

### Method of Partial Fractions

#### Definition: Partial Fractions

Rational expressions can be written as a sum of simpler fractions called partial fractions.

#### Definition: Distinct Linear Factors

Let $\displaystyle f(x)=\frac{P(x)}{Q(x)}$ be a rational function where\newline $deg(P) and $Q(x)$ is factorable into distinct linear factors. Then $\frac{P(x)}{Q(x)}=\frac{A}{a_1x+b_1}+\frac{B}{a_2x+b_2}+\cdots$

Let $\displaystyle f(x)=\frac{P(x)}{Q(x)}$ be a rational function where $deg(P) and $Q(x)$ is factorable into distinct quadratic factors. Then $\frac{P(x)}{Q(x)}=\frac{Ax+B}{a_1x^2+b_1}+\frac{Cx+D}{a_2x^2+b_2}+\cdots$

#### Definition: Repeated Linear Factor

Let $\displaystyle f(x)=\frac{P(x)}{Q(x)}$ be a rational function where\newline $deg(P) and $Q(x)$ is factorable into $r$-repeated linear factors, i.e., $Q(x)=(a_1x+b_1)^r$. Then

$\frac{P(x)}{Q(x)}=\frac{A}{a_1x+b_1}+\frac{B}{(a_1x+b_1)^2}+\frac{C}{(a_1x+b_1)^3}+\cdots+\frac{R}{(a_1x+b_1)^r}$

Let $\displaystyle f(x)=\frac{P(x)}{Q(x)}$ be a rational function where $deg(P) and $Q(x)$ is factorable into $r$-repeated quadratic factors, i.e., $Q(x)=(a_1x^2+b_1)^r$. Then

$\frac{P(x)}{Q(x)}=\frac{Ax+B}{a_1x^2+b_1}+\frac{Cx+D}{(a_1x^2+b_1)^2}+\frac{Ex+F}{(a_1x^2+b_1)^3}+\cdots+\frac{Sx+T}{(a_1x^2+b_1)^r}$

### Practice

Rewrite the following rational expressions into a sum of partial fractions:

1. $\displaystyle \frac{1}{x^2-3x-4}$
2. $\displaystyle \frac{x}{x^2-5x+4}$
3. $\displaystyle \frac{x-2}{x^2-9}$
4. $\displaystyle \frac{3x+2}{x^2-x}$
5. $\displaystyle \frac{x+1}{x^2-4}$
6. $\displaystyle \frac{3}{x^3+x}$
7. $\displaystyle \frac{2-x}{x^3-1}$
8. $\displaystyle \frac{x}{x^3+8}$
9. $\displaystyle \frac{x^2+x+1}{x^3-2x^2+x-2}$
10. $\displaystyle \frac{x-3}{x^4-16}$
11. $\displaystyle \frac{x^2+2}{x^5-3x^4+2x^3}$

#### Use Technology to Factor Expressions

Mathematica

1(* Practice 01: Factor x^2-3x-4 *)2Factor[x^2-3x-4]

#### Use Technology to Verify Partial Fractions

Mathematica

xxxxxxxxxx21(* Practice 01: Find partial fractions for 1/(x^2-3x-4) *)2Apart[1/(x^2-3x-4)]

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.