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 $$\frac{dy}{dt}= 0.06 y.$$

If the bank changes the rules, say by imposing a $\$ $100/year fee, then the equation changes to $$\frac{dy}{dt}= 0.06y - 100.$$

Population growth when there is plenty of food and space is modeled by the equation $$\frac{dy}{dt}=ky,$$ where $k$ is a constant.

If the food supply is limited, then this equation is modified to the logistic equation $$\frac{dy}{dt} = ky \left(1 - \frac{y}{M}\right).$$

Radioactive decay is governed by $$\frac{dy}{dt} = -ry.$$