Integrands must sometimes be factored before they can be separated. Cross multiplying and integrating is just a simple way to remember separation of variables. The relative growth rate is just the rate of change, or derivative. Every function with constant relative growth rate is an exponential function. A positive relative growth rate represents decay and a negative growth rate represents growth.
Compound interest has a constant relative growth rate.
The quantity of pollution in Lake Erie and the cooling of a body to room temperature are both examples of functions with stable equilibria. The differential equation for the velocity of a sky diver represents the net forces on the sky diver. The terminal velocity of a sky diver can be found without solving a differential equation. The quantity in the differential equation representing the salinity in a reservoir is the concentration, C. A differential equation must be solved explicitly before obtaining information about a function. The simplest model for growth rate, constant relative growth, leads to exponential growth.