TITLE: Extending the fundamental theorem of linear systems
QUESTION [0 upvotes]: Suppose we're given a system of equations
$x'(t) = A(t)x(t) + f(t)$, where $x$ and $f$ are vectors, and $A(t)$ acts as the analog of function-valued coefficients being comprised of general functions of $t$.
The one dimensional case $x'(t) = g(t)x(t) + f(t)$ admits a known solution in terms of the exponential and integrals of $g(t)$ and $f(t).$
But, I'm wondering if there exists a solution in the case that we instead have a system of equations, where $f(t)$ is a vector comprised of $(f_1(t),...,f_n(t))$ and $A(t)$ is a matrix of comprised of
$$\begin{bmatrix}
A_{1,1}(t) & A_{1,2}(t) & \cdots  \\
A_{2,1}(t) & A_{2,2}(t) & \cdots  \\
\vdots  & \vdots  & \ddots
\end{bmatrix}$$

REPLY [0 votes]: If we can find $n$ linearly independent solutions of the homogeneous system, say $\vec\varphi_1(t),\dots,\vec\varphi_n(t)$ and let
$$\mathbf{\Phi}(t)=(\vec\varphi_1(t),\dots,\vec\varphi_n(t)),$$
then we can express the solution as
$$\vec x(t)=\mathbf\Phi(t)\left(\vec C+\int_{t_0}^t\mathbf\Phi^{-1}(s)\vec f(s)\,ds\right).$$
However, in general, $\varphi_1(t),\dots,\varphi_n(t)$ are not easy to find. A special case that can be solved is that $\mathbf{A}(t)=
\mathbf{A}$ is constant. In this case, $\mathbf{\Phi}(t)=e^{\mathbf{A}t}$.
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