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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.

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    Here(http://en.wikipedia.org/wiki/Differential_algebra) but nothing elsea is a link to the Wikipedia page for differential algebra. I recalled the existence but nothing else of this field. Perhaps it will help.2012-07-08

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