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Lambert W function with rational polynomial

by nlooije
Tags: function, lambert, polynomial, rational
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Jul9-14, 05:28 PM
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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
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Jul9-14, 09:00 PM
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It doesnt show the steps but I got this from Wolfram
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Jul9-14, 09:53 PM
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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##.

Jul13-14, 07:30 AM
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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.

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