I'm struggling to understand why the reduction of order works. I'm reading the guide found here, and it starts with the assumption that, given a solution $y_1(t)$ we may be able to find another solution by multiplying with another function: $y_2(t) = v(t)y_1(t)$.
Previously, such derivations of other solutions, like the principle of superposition, were a result of the linearity of linear ODEs, but in this case, we are multiplying by a non-constant and no rules of linearity apply.
I still have a gut feeling that something relating to linearity is at play, but if this is not the case, then what property of second-order linear homogeneous ODEs allows us to make such an assumption?
Edit: In a linked section, the author justifies this assumption with
To find the second solution we will use the fact that a constant times a solution to a linear homogeneous differential equation is also a solution: $y_2(t) = v(t)y_1(t)$
And yet, the "constant" is another function? How can this be?