I think calling a DE "a fact about the derivative of a function" and then speaking about "solving it" in the same sentence is a big source for confusion, since we don't usually "solve facts".

EDIT: reading the whole thing, there's quite a bit of sloppiness and stuff that's flat-out wrong, like " the faster the cart goes, the faster it stops", or the statement at the end claiming that nonlinear DEs don't have solutions. In short: this needs a major cleanup.