Differential Equations

Equations Where the Unknown Is a Curve

A differential equation relates a quantity to its rate of change, and its solution is a function of time. Business laws are statements about rates: growth, decay, and adjustment speeds.

The king of them all:

$$\frac{dx}{dt} = r\,x \quad\Rightarrow\quad x(t) = x_0\,e^{rt}$$

The change is proportional to the current size. Deposits earning continuous interest or subscriber bases obeying it grow exponentially for $r > 0$ and decay for $r < 0$ (depreciation, churn).

Continuous compounding beats annual compounding at the same nominal rate: $e^r > 1 + r$.

Growth with a Ceiling

Markets saturate; nothing exponential survives contact with a finite world. The logistic equation adds a brake:

$$\frac{dx}{dt} = r\,x\left(1 - \frac{x}{K}\right)$$

where $K$ is the market capacity. Early on ($x \ll K$) it acts exponential; as $x \to K$ growth chokes to zero. The solution is the S-curve, the shape of every product's adoption.

The modeling discipline: Write the rate law you know, solve for the trajectory you want, then fit $x_0$ and $r$ to data. An ODE is a business assumption wearing mathematical clothes, laid bare for forecasting.

Tip: The doubling-time shortcut $t_2 = \ln 2 / r \approx 0.69/r$ is the cousin of the rule of 70. A process growing at 7% continuously doubles in about ten periods, no calculator needed.