Solution of linear ODE

Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$. It seems that the solution of linear ODE $$\dot x= Ax$$ can not be written in a closed form using $\int$, $A$, $x(0)$ and the standard functions.

Question 1. Is it a theorem?

If "no".

Question 2. Any idea how to prove such statement?

• @fedja and Co: I do not see how to make it more clear... – ε-δ Sep 4 '13 at 19:24

This is known as Differential Galois Theory, first developed by Picard and Vessiot. In your case you should look for authors such as Kolchin or Singer and Van Der Put. Some systems definitely admit solutions in "closed form" (you can build them!), but most won't. The ingredient is the "monodromy group", measuring the multivaluedness of the analytic continuation (in the complex line) of local solutions, whose Zariski closure is an algebraic subgroup of $GL(n,\mathbb C)$ (assuming your equation starts with coefficient in $\mathbb R$ or $\mathbb C$). The solvability of the (connected component of the identity) of this algebraic group is the criterion for solvability in "closed form".

A comprehensive reference is:

van der Put, Marius; Singer, Michael F

"Galois theory of linear differential equations"

Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328. Springer-Verlag, Berlin, 2003.

MathSciNet : MR1960772

• Thank you very much. Could you find specific reference for me? – ε-δ Sep 4 '13 at 19:25
• As I told you, the book by Singer and Van der Put is very complete (you can actually do this for any differential field, not only $\mathbb C((x))$. – Loïc Teyssier Sep 4 '13 at 21:23