I have an equation I have been playing with (the variable is $x$ and the constants are positive and real):
$$( r_1 - x )^2 \frac{ a r_3 e^{x / r_2} + b r_2 e^{x / r_3} }{\left( a e^{ x / r_2} + b e^{ x / r_3} \right)^2} = k.$$
Certain approximations can lead to an equation that looked better to me.
It has the form $e^{2 x} +ae^x = bx +c $ where the symbols no longer have the same meaning. Without the $e^{2x}$ term I can find the solution easily with Lambert's W and it might be good enough for my purposes.
Does anyone have experience with equations like these? Do you know of a resource that might show lots of examples of the W function in action? Or, do you have a reason why an analytical solution here is not reasonable and I should just settle for the numerical approach?
What branch of advanced algebra would govern these kinds of problems? Any tips on keywords for my research? Like I said I'm a physicist and don't necessarily know the math search terms to use.