Here are several systems described by first-order differential equations. In each case there is a "law", or a set of "rules of the game", that describe how the system changes with time.
Examples:
The law of banks: If you invest money in a bank paying 6% interest, then the amount y(t) of money that you have at time t obey dydt=0.06y.
If the bank changes the rules, say by imposing a $100/year fee, then the equation changes to dydt=0.06y−100.
Population growth when there is plenty of food and space is modeled by the equation dydt=ky,
where k is a constant.
If the food supply is limited, then this equation is modified to the logistic equation dydt=ky(1−yM).