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Lambert W function with rational polynomial 
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#1
Jul914, 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 


#2
Jul914, 09:00 PM

P: 20

It doesnt show the steps but I got this from Wolfram



#4
Jul1314, 07:30 AM

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P: 39,302

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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