TITLE: Floquet Theory - Reducing ODE's to Constant Coefficient ODE's?
QUESTION [10 upvotes]: Am I right in my reading of the bottom of this page by Arnold, where he apparently says that to any system of first order ODE's with periodic coefficients one can find a change of variables reducing the system to one with constant coefficients? 
In other words, we can find a change of variables that reduces something like
$$y''+\sin(x)y'+\cos(x)y=0$$
to a constant coefficient ODE by Floquet's theorem? Surely this isn't right is it? Should I be as shocked with this as I am?
Can one do this if we similarly start throwing in crazier periodic functions as coefficients, like saw-tooth functions etc...?

REPLY [2 votes]: This just a comment
I just stumbled over this question. The specific differential equation you mentioned 
\begin{equation}
y''+\sin(x)y'+\cos(x)y=0\hspace{2cm}(1)
\end{equation}
can be solved exactly without Floquet theory. It can be rewritten as
$$
y''+\frac{d}{dx}[\sin(x)y]=0\implies y'(x)+\sin(x)y=c
$$
If you set the initial values at $x=0$, for example: $y(0)=0$ and $y'(0)=1$, then
$$
y'(x)+\sin(x)y=y'(0)=1
$$ 
Now multiply this by an integrating factor $A(x)$:
$$
A(x)y'(x)+A(x)\sin(x)y=A(x)
$$
Taking $A'(x)=\sin(x)A(x)$, we can integrate this equation and obtain 
$$
A(x)=A_0e^{-\cos(x)+1}
$$
Hence, you find
$$
\frac{d}{dx}[A(x)y(x)]=A(x),
$$
which can be solved. You find
$$
y(x)=e^{\cos(x)}\int_0^x e^{-\cos(x')}dx'
$$
One can then expand the above solution in Fourier series (with some Bessel function coefficients) and compare it with the Floquet theory solution. You can then find the
Floquet multipliers. Further, this solution is the Green's function for  equation (1).
Consequently, you can also solve equations like these
$$
y''+\sin(x)y'+\cos(x)y=f(x)
$$