What's the general method for solving differential equations of this type
$$\frac d{dt} \pmatrix {y \\ x} = \pmatrix{y_0 \\ x_0} + y\pmatrix{a \\ b}$$
where $y=y(t), x=x(t)$, and $y_0$, $x_0$, $a$, and $b$ are real constants?
Should I just rewrite this as a second order scalar differential equation or is there some way of solving it in this vector form?
I'd shift into scalar form. You can solve $$\frac{dy}{dt}=y_0+ay$$ first, then use that solution to solve $$\frac{dx}{dt}=x_0+by$$ readily.
Alternately, you could start by solving the homogeneous equation $$\frac{d}{dt}\begin{pmatrix}y\\x\end{pmatrix}=\begin{pmatrix}a & 0\\b & 0\end{pmatrix}\begin{pmatrix}y\\x\end{pmatrix},$$ then proceed from there to the (potentially) nonhomogeneous case.