Note on a well-known equation in cosmological perturbation theory which is in error

We have a well-known equation in cosmological perturbation theory which appeared only by several simple algebraic errors made in many textbooks. There have been attempts to modify Newtonian equations aiming to reproduce that incorrect equation. We clarify why such attempts are wrong, present the correct equation to try in the modification, and explain its own limitation as well. We show that any form of density perturbation equation is possible by a suitable gauge condition.