TITLE: Periodic Solutions for the system
QUESTION [5 upvotes]: I'm having lots of troubles with proving that systems do not have periodic solutions. For example, 
PROBLEM:
Prove that the system:
$\dot x = A(t)x$ has no periodic solutions but the trivial, where
$A(t) = \left [ \begin{matrix}
    0 & 2+sin(t) \\
    4+cos^{2}(t) & 0 \\
   \end{matrix} \right ]$
My naive attempt has to try to calculate the solutions explicitly but I went no where. My other idea is find some criterion for perturbed systems (i.e $\dot x = (A+B(t))x$) which allows me to prove periodicity but I couldn't find any. 
Any help would be incredibly appreciated, thanks guys <3

REPLY [6 votes]: For your system, 
$$ \eqalign{\dfrac{d}{dt} (x(t) y(t)) &= \dot{x}(t) y(t) + x(t) \dot{y}(t)\cr & = 
  x \left( t \right)  ^{2} \left( 4+ \left( \cos \left( 
t \right)  \right) ^{2} \right) +  y \left( t \right)  ^
{2} \left( 2+\sin \left( t \right)  \right)\cr 
&> 0}$$ as long as $(x(t), y(t)) \ne (0,0)$.