I am studying equations of the form
$f_1 e^{g_1z}+f_2e^{g_2z}=f_3e^{g_3z}$
where the $f$'s and $g$'s are functions of the form $ax+by$, with $a, b \in \mathbb{R}$. I'm interested in things like
$2xe^{2xz}+4ye^{(3x+y)z}=(2x+4y)e^{yz}$
I feel like there should be a straightforward way to solve/simplify this given that $e$ in there, but I can't figure it out. Maybe I'm missing something simple. Or can equations like this really be solved only via numerical means? Are there some advanced techniques I could consider to get an analytic solution (in terms of x, y, z)?
Thank you
If $z$ is your unknown, even
$$e^{5z}+e^z+1=0$$ can't be solved analytically (it is equivalent to $t^5+t+1=0$).
If $x$ is your unknown, even
$$xe^x=1$$ cannot be solved, but with the help of Lambert's $W$ special function.
There is no hope of an analytic solution in the general case, this equation family is "highly" transcendental.