Consider the matrix ode $$\frac{\mathrm d x}{\mathrm d t}(t)=x(t)A(t),x(0)=c$$where $x(t)\in\mathbb R^n, A(t) \in M_{n\times n}$ is continuous. It seems that we can use perturbative method (which I'm not sure if this is reasonable) to solve the matrix equaiton $Y_{\epsilon}(t)\in GL_n$ $$Y'_{\epsilon}(t)=\epsilon Y_{\epsilon}(t)A(t),Y_{\epsilon}(0)=I$$ where $Y_{\epsilon}(t)=I+Y_1(t)\epsilon+Y_2(t)\epsilon^2+\cdots$ In the end we get the following formula $$x(t)=cY(t),Y(t)=I+\int_{0}^tA(t_1)dt_1+\int_0^t[\int_0^{t_2}A(t_1)dt_1]A(t_2)dt_2+\cdots\tag{1}$$ (a) In the Wikipedia page wiki it says in general there is no closed form solution except the method Magnus expansion using iterated commutators. I cannot see how Magnus expansion related to (1). Does iterated integral (1) has a name in ODE theory? Any reference or comment is appreciated!
(b) It seems that the perturbative method here can be generalized as follows, consider $Y(t)\in C^{\infty}(\mathbb R)\otimes \mathcal A$ where $\mathcal A$ is a completed graded algebra, while $A(t)\in C^{\infty}(\mathbb R)\otimes \mathcal A_1$, thus we can write homogeneous expansion $I(t)=\sum_0^{\infty}I_m(t)$ then the equaiton(the derivative here is point wise derivative $\frac{d}{dt}\otimes \mathrm{id}$) $$Y'(t)=Y(t)A(t)$$ can also be solved as (1)! Is there a general context of this?