I was trying to solve the following equation for t
$$(P\cdot l \cdot \exp(-l\cdot t) + R \cdot l \cdot \exp(-l \cdot t))/t + (P \cdot \exp(-l \cdot t) + R \cdot (\exp(-l \cdot t) - 1))/t^2 = 0 $$
Being a little bit nasty, I tried to put this equation on MATLAB, but it says "Explicit solution could not be found".
Is it easy to see numerically that the solution actually exists.
So my question is: 1) why the solution cannot be found? 2) is there any criteria to check solvability of algebric equations? (any external source link is welcomed) 3) can I find any approximation of the solution?
Thank you very much.
EDIT: P, R and l are strictly positive.
It is not possible to solve the equation in terms of a finite number of elementary functions. But it is possible thanks to a special function: http://mathworld.wolfram.com/LambertW-Function.html