What's wrong with this method of solving linear ODE systems with time varying coefficients? The problem is stated: "What is wrong with the following calculation for an arbitrary continuous matrix $A(t)$?
$$\frac{d}{dt}\big[\exp \int_{t_0}^t A(s) ds\big] = A(t) \exp \big[ \int_{t_0}^t A(s) ds \big] $$
so that $\exp (\int_{t_0}^t A(s) ds)$ is a fundamental matrix of $\mathbf{y}' = A(t)\mathbf{y}$ for an arbitrary continuous matrix $A(t)$."
This seems like a straightforward application of the chain rule and the fundamental theorem of calculus, so I'm not sure what the question-writer is looking for.
 A: The basic problem is that $A(t)$ and $\int_{t_0}^t A(s) \; ds$ won't commute for general $A(t)$:
$\left [A(t), \displaystyle \int_{t_0}^t A(s) \; ds \right ] \ne 0; \tag 1$
this issue manifests itself when forming
$\dfrac{d}{dt} \left [ \exp \left ( \displaystyle \int_{t_0}^t A(s) \; ds \right ) \right ]; \tag 2$
with the matrix exponential defined by the series expansion
$\exp \left (\displaystyle \int_{t_0}^t A(s) \; ds \right ) = \displaystyle \sum_0^\infty \dfrac{1}{n!} \left ( \displaystyle \int_{t_0}^t A(s) \; ds \right )^n, \tag 3$
we encounter difficulties with the terms of degree $2$ and higher, as is illustrated by
$\dfrac{d}{dt} \left ( \displaystyle \int_{t_0}^t A(s) \; ds \right )^2 = A(t)  \left ( \displaystyle \int_{t_0}^t A(s) \; ds \right )  + \left ( \displaystyle \int_{t_0}^t A(s) \; ds \right ) A(t); \tag 4$
in the light of (1) we cannot interchange the factors in the second term to bring $A(t)$ to the forefront, and the same issue evidently pertains to every power of the integral occurring in the sum on the right of (3); and without this commutation, there is no way to validate
$\dfrac{d}{dt} \left [ \exp \left ( \displaystyle \int_{t_0}^t A(s) \; ds \right ) \right ] = A(t) \exp \left ( \displaystyle \int_{t_0}^t A(s) \; ds \right ) \tag 5$
as there is in the case of a one-dimensional variable $a(t)$, for which
$\dfrac{d}{dt} e^{a(t)} = a'(t) e^{a(t)} \tag 6$
follows easily from the ordinary chain rule of single-variable calculus.
For matrices $A(t)$ such that 
$\left [A(t), \displaystyle \int_{t_0}^t A(s) \; ds \right ] \ne 0 \tag 7$
for all $t$ and $t_0$, however, the formula (5) applies.  Once class of such matrices which has some proven utility is 
$A(t) = f(t) B, \tag 8$
where $B$ is a constant matrix; it is easy to see that (5) binds for such $A(t)$.
Nota Bene: I have encountered this situation so many times in my own work with time-dependent, linear ordinary differential equations that I can only say I wish (5) were true; it would certainly make many things hella' easier! End of Note.
