Register to reply

Lambert W function with rational polynomial

by nlooije
Tags: function, lambert, polynomial, rational
Share this thread:
nlooije
#1
Jul9-14, 05:28 PM
P: 1
Hi all,

During my research i ran into the following general type of equation: [itex] \exp(ax+b)=\frac{cx+d}{ex+f}[/itex]
does anyone have an idea how to go about solving this equation?

thx in advance
Phys.Org News Partner Mathematics news on Phys.org
Math journal puts Rauzy fractcal image on the cover
Heat distributions help researchers to understand curved space
Professor quantifies how 'one thing leads to another'
Ledsnyder
#2
Jul9-14, 09:00 PM
P: 26
It doesnt show the steps but I got this from Wolfram
Attached Thumbnails
atrttr.gif  
micromass
#3
Jul9-14, 09:53 PM
Mentor
micromass's Avatar
P: 18,346
Quote Quote by Ledsnyder View Post
It doesnt show the steps but I got this from Wolfram
I think the idea is to solve for ##x## instead of ##f##.

HallsofIvy
#4
Jul13-14, 07:30 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682
Lambert W function with rational polynomial

Let [itex]u= \frac{cx+ d}{ex+ f}[/itex], the fraction on the right. Then, solving for [itex]x[/itex], [itex]x= \frac{d- fu}{eu- c}= -\frac{f}{e}u+ \frac{fc}{e}[/itex].

So the equation is, so far,
[tex]e^{ax+ b}= e^{-\frac{af}{e}u+ \frac{afc}{e}+ b}= u[/tex]
[tex]e^{-\frac{af}{e}u}e^{\frac{afc+ bd}{d}}= u[/tex]
[tex]ue^{\frac{af}{e}u}= e^{\frac{afc+ bd}{d}}[/tex]

Let [itex]v= \frac{af}{e}u[/itex]. Then [itex]u= \frac{e}{af}v[/itex] and we have
[tex]\frac{e}{af}ve^v= e^{\frac{afc+ bd}{d}}[/tex]
[tex]ve^v= \frac{af(af+ bd)}{de}[/tex]

[tex]v= W(\frac{af(af+ bd)}{de}[/tex]

Now work back through the substitutions to find x.


Register to reply

Related Discussions
Lambert W function General Math 2
Simplification of a rational polynomial function General Math 4
Using lambert's w function Calculus & Beyond Homework 0
Limit of rational function to rational power Calculus & Beyond Homework 4
Limits of a Polynomial/Rational Function General Math 7