# linear first order ODE with polynomial coeffficients

I'm wondering if anything can be said about the solution to a system of linear first order ODEs with polynomial coefficients, especially about analyticity of the solution.

The equation is given as $\dot{x}=A(t)x$ where $A(t)$ is a matrix polynomial in t.

Could we conclude that the solution $x$ is analyitic?

Note: it is not to be assumed that $A(t_1)$ commutes with $A(t_2)$ for any $t_1\neq t_2$.

In a previous post Finding Weak Solutions to ODEs, it was commented that this can be done by Fourier Transformation.

Could anybody help me with that? Thanks!