A book I'm using to teach myself differential equations claims the following:
If $y_{1}$ and $y_{2}$ are solutions to the differential equation $y' - a(t)y = q(t)$, then $y = y_{1} - y_{2}$ will be a null solution by linearity.
I understand there exists some linear combination of $y_{1}$ and $y_{2}$ that would provide the null solution, but how can I be sure it is exactly $y = y_{1} - y_{2}$?
You can check this by hand:
Suppose $y_1$ and $y_2$ are solutions, so that
$y_1' - a(t) y_1 = q(t)$ and $y_2'- a(t) y_2 = q(t).$
Then, subtracting these two equations yields
$$y_1'-y_2' - a(t) y_1 +a(t) y_2 = 0.$$
A slight rearranging shows
$$(y_1-y_2)' - a(t) (y_1-y_2) = 0,$$
so indeed, $y=y_1-y_2$ is a null solution.