I do not know topology (noob ) but I am sure if those "drop of terms" is proved. So it is okay.

Here is what I am reffering to when I say approximation,

It is known that a swinging pendulum satisfies (am I right?):

sin y+y'=f(t)

What physcisits do is approximate the solution for small t that is sin t=t then,

y'+y=f(t)

This is a linear differencial equation and it becomes simpler.

However, a mathematicians would not be satisfied with such an approach because it is not exact that t=sin t

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To add I think there is something called "linearizing" in applied math to simplify the functions but mathematicians do not deal with such problems. Even more they are useless.