Can we find a Taylor Series expansion for $y(x)$ implicitly defined by: $$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ?$$

In financial mathematics, the two-additive-factors Model G2++ is commonly used for interest rates forecasting. G2++ Model parameters are calibrated on market Caps and Swaptions prices. On the basis of these calibrated parameters complex financial product prices are computed using Monte-Carlo methods.

For further details on G2++ models refer for example to Brigo & Mercurio

Numerical computational challenges arise due to the necessity to integrate an implicitly defined function of the form:

$$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ,$$

which implicitly defines $y(x)$. All other parameters $A_i$, $a_i$ and $b_i$ are constant positive real numbers.

Now comes my question: What is the Taylor of $y(x)$ around $x=0$?

I started with a highly simplified version of the equation:

$$e^{x+y}+e^{x+by} = 1,$$

And found explicit closed form solutions for $y(x)$ when $b \in \{0,1,2,3,4\}$. This corresponds to the cases when the equation becomes Galois solvable. But what if $b$ is irrational, or arbitrarily large?

At least for this simplified single parametric version, wouldn't there be a strategy to obtain Taylor Series expansions?

Thanks for your answers or suggestions! They are greatly appreciated.

Looking at the case $b=2$, it seems that $y(x)$ is really close to its linear asymptotes $x+y=0$ and $x+by=0$. maybe a clue? see here.