Differential Equations

Equations Where the Unknown Is a Curve

A differential equation relates a quantity to its own rate of change — and its solution is not a number but a function of time. Business is full of them, because most business laws are statements about rates: growth rates, decay rates, adjustment speeds.

The king of them all:

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

'The change is proportional to the current size.' Deposits earning continuous interest, subscriber bases growing by word of mouth, debts compounding: all obey it, growing exponentially for $r > 0$ and decaying for $r < 0$ (depreciation, brand recall fading, radioactive inventory of relevance).

Reading the solution:

• $x_0$ is the initial condition — the same ODE with different starting points draws parallel destinies.
• Doubling time: $t_2 = \ln 2 / r \approx 70/(100r)$ — the famous rule of 70.
• Continuous compounding at rate $r$ beats annual compounding at the same nominal rate: $e^r > 1 + r$.

One step richer — growth with a ceiling. Markets saturate; nothing exponential survives contact with a finite world. The logistic equation adds the brake:

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

where $K$ is the market's capacity. Early on ($x \ll K$) it looks exponential; as $x \to K$ growth chokes to zero. The solution is the S-curve — the shape of every successful product's adoption: slow start, explosive middle, saturated plateau.

The modeling discipline: write what you know (the rate law), solve for what you want (the trajectory), then fit $x_0$ and $r$ to data. An ODE is a business assumption wearing mathematical clothes — and the solution curve is that assumption's forecast, laid bare for checking.

Reading $x(t) = x_0 e^{rt}$

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