Solution to non-autonomous system of ODEs I have the following system of ordinary differential equations with $\gamma(t),M(t),$ and $\beta(t)$ strictly positive and smooth.
$\dot{g_1}(t) = \gamma(t)M(t){g_2}(t)$
$\dot{g_2}(t) = \beta(t)\frac{1}{M(t)}{g_1}(t)$
with initial conditions
${g_1}(0)=1$
${g_2}(0)=1$
$0\le t \le 1$
How would I go about finding a non-trivial solution?
Editing to add that $\gamma(t), M(t),$ and $\beta(t)$ are finite on $[0,1]$
2nd edit: Corrected typo in domain of $t$
 A: EEDDIITT: given a system $$ f_1' = g f_2, \; \; \; f_2' = h f_2,  $$ you get separate equations
$$   f_1'' = \left( \frac{g'}{g} \right) f_1' + g h f_1  $$
$$   f_2'' = \left( \frac{h'}{h} \right) f_2' + g h f_2.  $$
ORIGINAL: Currently see no hope of a closed form solution in general. Bounds are available, and numerical solution always is, assuming you have your coefficient functions in an explicit manner. 
There is one circumstance with a solution, a fairly restrictive one. Let
$$  A(t) = \int_0^t \gamma(s) M(s) ds, $$
$$  B(t) = \int_0^t \frac{\beta(s)}{ M(s)} ds. $$
The restrictive condition is
$$   \frac{A}{B}  $$ constant. In that case, the fundamental solution matrix is
$$
\left(
\begin{array}{cc}
\cosh \sqrt {AB} & \frac{\sqrt A \sinh \sqrt {AB}}{\sqrt B} \\
\frac{\sqrt B \sinh \sqrt {AB}}{\sqrt A} & \cosh \sqrt {AB}
\end{array}
\right).
$$
With your initial condition,
$$ g_1(t) = \cosh \sqrt {AB} + \frac{\sqrt A \sinh \sqrt {AB}}{\sqrt B},  $$
$$ g_2(t) = \cosh \sqrt {AB} + \frac{\sqrt B \sinh \sqrt {AB}}{\sqrt A}.  $$
The theorem used here is that, for a square matrix $H(t),$ if $H$ and $\dot{H}$ commute, then
$$ \frac{d}{dt} e^H = e^H \dot{H } = \dot{H} e^H.  $$
