If we have a first order linear differential equation $q(t)y'(t) + k(t)y(t) = p(t)$, we can take $q(t)$ to not be identically zero, because otherwise we don't even have a differential equation. Therefore, we often divide through by $q(t)$ to get $y'(t) + g(t)y(t) = u(t)$ where $g = \frac{k}{q}$ and $u = \frac{p}{q}$, and we usually base our study of first-order linear ODEs from this simplified equation.
However, while we can assume $q(t)$ to not be identically zero, there is still also the case that $q(t)$ is not identically zero, but it is sometimes $0$ for certain values of $t$. Hence dividing by $q(t)$ is still not safe.
Is there an explanation for why this is allowed? Or is this just sloppiness we allow ourselves to get away with?
Note this also applies to higher-order linear ODEs, where we generally ignore the coefficient of largest-order $y^{(n)}$ by dividing through.