I have encountered a problem in my elec. eng research that I find rather challenging, being a bear of very little brain. The question has been considered under a slightly different aspect here: <http://arxiv.org/abs/math-ph/0607011>


My formulation of the problem is as follows: with all quantities real, is it possible to express the smallest positive $x$ s.t.


$e^{ax}-b(x-r\_1)(x-r\_2)=0$


in closed form? The Lambert W function easily solves the case with one factor $(x-r)$. 


Returning to the "quadratic" case, depending on the sign of $b$ one can have 0,1,2 or even three real solutions, the "2+" cases having a clear connection with the two real branches of W (at least in approximation), expressing two solutions "on either side" of $r\_1$ (or $r\_2$). My specific application constrains $a<0$ and I usually consider $r\_1\lt r\_2$ with no further restrictions on the sign of either.


My hope is that someone on this list might find the question entertaining in a more general context; perhaps indicate or preclude a standard method of attack. Numerical solutions are straightforward, but I wonder if more physical insight might be achieved otherwise. 



An interesting equation of some practical interest.
You might be interested in a series solution. After a change of variable we can rewrite the equation in the form
$$ {{\rm e}^{-x}}-b x \left( sx-1 \right) = 0$$
so that for $s=0$ we have a solution at $x = w$ where $w = \text{LambertW}(-1/b)$.
Then we have a series solution in powers of $s$, which should converge for small $|s|$:


$$\eqalign{ x &= w+{\frac {{w}^{2}s}{w+1}}+{\frac {{w}^{3} \left( w+2 \right) ^{2}
{s}^{2}}{2 \left( w+1 \right) ^{3}}}+{\frac {{w}^{4} \left( 2{w}
^{4}+17{w}^{3}+48{w}^{2}+60w+30 \right) {s}^{3}}{6 \left( w+1
 \right) ^{5}}}\cr &+{\frac {{w}^{5} \left( w+2 \right) \left( 6{w
}^{5}+68{w}^{4}+257{w}^{3}+462{w}^{2}+420w+168 \right) {s}^{4}
}{ 24\left( w+1 \right) ^{7}}}\cr&+{\frac {{w}^{6} \left( 
24{w}^{8}+442{w}^{7}+3172{w}^{6}+12149{w}^{5}+28040{w}^{4}+
40860{w}^{3}+37440{w}^{2}+20160w+5040 \right) {s}^{5}}{ 120 \left( w
+1 \right) ^{9}}}\cr
&+\ldots}
$$


