There are some methods for solving a class of non-linear systems, so-called conditionally linear which posit the following form:
$$\dot y_i = a(y_{\sim i}) y_i +b(y_{\sim i})\hspace{1cm} \forall i={1,...,d}$$
for the $i$th component of a a $d$-dimensional system. The key here is that the system can be written in a linear form if the non-lienar functions $a$ and $b$ are only functions of components $j \ne i$ (denoted by $\sim i$ here).
It seems to me that, at least in a large range of problems, it is possible to separate a linear term from the non-linear functions treat the system as if it was conditionally linear. For instance, consider
$$\dot x = xy \sin(-y)$$ $$\dot y = y^3 \exp(-x)$$
which can be cast as
$$\dot x =x \times F(y)$$ $$\dot y =y \times G(x,y)$$
with
$$F(y) = y\sin(-y)$$ $$G(x,y) = y^2\exp(-x)$$
Clearly, while $F$ is $x$-independent, $G$ is not $y$-independent. So strickly speaking, the system is not conditionally linear. However, there are numerical methods that treat this system like so, by using implicit-explicit methods. For instance, one can numerically construct the rates at time step $n$, by using the values of $(x,y)$ at the previous time steps (explicit):
$$\dot x_n =x_n \times F(y_{n-1})$$ $$\dot y_n =y_n \times G(x_{n-1},y_{n-1})$$
and solve the system above using an implicit technique. In this numerical setting, the system looks like a conditionally linear system, although the continuous one is not.
My question is whether the discretized system is a conditionally linear one. And if so, can I use numerical methods such as exponential Euler, as suggested by Mascagni & Sherman and get the same convergence rate?