# Lambert W function with rational polynomial

by nlooije
Tags: function, lambert, polynomial, rational
 P: 1 Hi all, During my research i ran into the following general type of equation: $\exp(ax+b)=\frac{cx+d}{ex+f}$ does anyone have an idea how to go about solving this equation? thx in advance
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,302 Lambert W function with rational polynomial Let $u= \frac{cx+ d}{ex+ f}$, the fraction on the right. Then, solving for $x$, $x= \frac{d- fu}{eu- c}= -\frac{f}{e}u+ \frac{fc}{e}$. So the equation is, so far, $$e^{ax+ b}= e^{-\frac{af}{e}u+ \frac{afc}{e}+ b}= u$$ $$e^{-\frac{af}{e}u}e^{\frac{afc+ bd}{d}}= u$$ $$ue^{\frac{af}{e}u}= e^{\frac{afc+ bd}{d}}$$ Let $v= \frac{af}{e}u$. Then $u= \frac{e}{af}v$ and we have $$\frac{e}{af}ve^v= e^{\frac{afc+ bd}{d}}$$ $$ve^v= \frac{af(af+ bd)}{de}$$ $$v= W(\frac{af(af+ bd)}{de}$$ Now work back through the substitutions to find x.