Consider the following sytem of ODEs
$\dot{x}= Ax$, and given $x(0)$,
where $A$ is a $n\times n$ matrix with rational entries.
Can I encode the solution, say $x(t)$ for a given $t$, as a first order logic formula over $(R, +, \cdot, e^x, 0, 1)$?
Here, let us assume that $x(t)$ has real entries.
The difficulty I can see is that some eigenvalue of A is an algebraic number, so the solution invloes some sin and cosin. I do not know how to overcome this.
Thanks.