Solution of $U'(t) = (A + tB)U(t)$ Is there a known general solution to a differential equation of the form
$$U'(t) = (A + tB)U(t),$$
where $A$ and $B$ are constant $n \times n$ matrices and $U$ is a function mapping $\mathbb{R}$ to $\mathbb{R}^n$ ?
 A: Yes and no, it depends. Let's define $X := tA + \frac{1}{2}t^2B$, such that $\dot{X} = A+tB$ (because $A$ and $B$ are constant matrices) and $\dot{U} = \dot{X}U$. As discussed on this page, this equation can be solved in the same way as scalar ODEs, i.e. with the solution $U(t) = e^X$, on the condition that $X$ observes the relation $\displaystyle\sum_{n=1}^\infty\frac{[X,\dot{X}]_n}{(n+1)!} = 0$, where $[A,B]_n = [A,[A,[A,\ldots]]]$ is the $n$-time nested commutator; this condition is trivially satisfied in the case $[X,\dot{X}] = \frac{1}{2}t^2[A,B] = 0$, in other words, when $A$ and $B$ commute.
If the aforementioned conditions are not met, you can still integrate recursively the differential equation, as follows :
$$
\begin{array}{rcl}
U(t) 
   &=& \displaystyle
   U(t_0) + \int_{t_0}^t(A+\tau B)U(\tau)\mathrm{d}\tau \\
   &=& \displaystyle
   U(t_0) + \int_{t_0}^t\mathrm{d}\tau_1\,(A+\tau_1B)U(t_0) + \int_{t_0}^t\mathrm{d}\tau_1\int_{t_0}^{\tau_1}\mathrm{d}\tau_2\,(A+\tau_1B)(A+\tau_2B)U(\tau_2) \\
   &=& \displaystyle
   \ldots \\
   &=& \displaystyle
   \sum_{n=0}^\infty U_n(t)U(t_0)
\end{array}
$$
where
$$
U_n(t) := \int_{t_0}^t\mathrm{d}\tau_1\int_{t_0}^{\tau_1}\mathrm{d}\tau_2\cdots\int_{t_0}^{\tau_{n-1}}\mathrm{d}\tau_n\, V(\tau_1) \cdots V(\tau_n) \quad\mathrm{with}\quad V(t) := A+tB
$$
This form can be seen as a series expansion of the solution, which is named "Dyson series" in physics (see here); thanks to a reparametrization "trick", it can even be reformulated in a more customary form, namely :
$$
U(t) = \mathcal{T}\exp\left(\int_{t_0}^tV(\tau)\,\mathrm{d}\tau\right)U(t_0)
$$
where $\mathcal{T}$ is the so-called "time-ordering operator". In the case where you are only interested in a "small-time" approximated solution, you can limit yourself to the first-order expansion for instance, that is
$$
U(t) \approx  U(t_0) + \int_{t_0}^t(A+\tau B)U(t_0)\mathrm{d}\tau = \left(1+(t-t_0)A+\frac{1}{2}(t-t_0)^2B\right)U(t_0)
$$
