Partial answer. This seems to be equivalent to asking whether you chosen language can express the notion "$n$ is an integer".
First, if you can express solutions to linear ODEs, then an "integer" is a number $t$ such that $\sin(\pi t)=0$, and $(\sin,\cos)$ are solutions to a well-known ODE system of the kind you're considering.
On the other hand, if we can express integerness and also have $\cdot$ and $+$, then every computable function can be expressed (using Gödel's $\beta$ function to encode primitive recursion and building up from there). And then we can just program a numerical simulation of the ODE and write down a first-order-formula for "the limit of the output of the simulation as the step size tends to 0".
So can integers be recognized by $(\mathbb R, {+}, {\cdot}, \exp, 0, 1)$? Beats me. It's not obviously possible, but on the other hand the real exponential function is closely enough related to the complex one that I won't rule it out either.