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 $deg(P)<deg(Q)$ and $Q(x)$ is factorable into distinct linear factors. Then $\frac{P(x)}{Q(x)}=\frac{A}{a_1x+b_1}+\frac{B}{a_2x+b_2}+\cdots$

Definition: Distinct Quadratic Factors

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

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

Definition: Repeated Quadratic Factors

Let $\displaystyle f(x)=\frac{P(x)}{Q(x)}$ be a rational function where $deg(P)<deg(Q)$ 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 *)
2
Factor[x^2-3x-4]

Use Technology to Verify Partial Fractions

Mathematica

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

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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 18:39 EDT