# Differential Equations

Business I · Mathematics I · https://tryals.app/learn/business-i/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).

| Component | Business Meaning |
| --- | --- |
| $x_0$ | Initial condition, first cohort |
| $r > 0$ | Compound growth, viral adoption |
| $r < 0$ | Decay, depreciation, churn |
| $\ln 2 / r$ | Doubling or halving time |

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.

## Practice questions

8 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Which statements about $x(t) = x_0 e^{rt}$ are true?

A. Its doubling time depends on the current size $x$
B. Continuous compounding at rate $r$ yields more than simple annual $r$
C. It solves $\dot{x} = rx$ for any initial value $x_0$
D. With $r < 0$ it decays toward zero without ever crossing it

**Answer:** B. Continuous compounding at rate $r$ yields more than simple annual $r$; C. It solves $\dot{x} = rx$ for any initial value $x_0$; D. With $r < 0$ it decays toward zero without ever crossing it

**Why:** Doubling time $\ln 2/r$ is the exponential’s magic: *independent of size*. A million doubles as fast as a hundred, which is why compounding stories end so dramatically.

Page: https://tryals.app/practice/business-i/differential-equations/which-statements-about-x-t-x0-e-rt-are-true

### 2. Equipment worth 80k€ depreciates continuously at $r = -10\%$/year. Set the slider to its approximate value after 7 years (k€). Note $e^{-0.7} \approx 0.5$.

**Answer:** 40 (within ±3)

**Why:** $80 \cdot e^{-0.7} \approx 80 \cdot 0.5 = 40$k€. The rule of 70 works for decay too: at 10% the half-life is about 7 years, depreciation schedules are radioactive decay wearing a suit.

Page: https://tryals.app/practice/business-i/differential-equations/equipment-worth-80k-depreciates-continuously-at-r-10-year-set

### 3. Exponential growth and logistic growth agree closely at first and diverge completely later. What in the equation causes the split?

A. The logistic model includes random noise that the exponential model omits
B. Exponential growth is discrete while logistic growth is continuous
C. The logistic growth rate falls near a carrying capacity
D. The logistic model uses a smaller growth constant throughout

**Answer:** C. The logistic growth rate falls near a carrying capacity

**Why:** The (1 − N/K) factor is nearly 1 when N is small and heads to zero near the ceiling, so the two models are indistinguishable exactly when a startup is fitting its curve. That is why early growth is such weak evidence about which regime you are in.

Page: https://tryals.app/practice/business-i/differential-equations/exponential-growth-and-logistic-growth-agree-closely-at-first-and

### 4. Logistic growth adds a carrying capacity $K$ to exponential growth. Why does that change the long-run forecast so drastically when both start identically?

A. Because the logistic curve grows faster at first and then reverses
B. Because exponential models are only valid for populations, not markets
C. Because $K$ changes the initial condition rather than the dynamics
D. Because the growth rate is scaled by how much room is left

**Answer:** D. Because the growth rate is scaled by how much room is left

**Why:** Both fit early data well, which is what makes the choice dangerous: the model, not the data, is doing the forecasting once the ceiling starts to bite.

Page: https://tryals.app/practice/business-i/differential-equations/logistic-growth-adds-a-carrying-capacity-k-to-exponential-growth-why

### 5. Two deposits obey the same ODE $\dot{x} = 0.05x$ but start at different amounts. Their balance curves will eventually cross.

**Answer:** False

**Why:** Solutions of the same first-order ODE through different initial conditions never cross: both are $x_0 e^{0.05t}$ with different $x_0$, and their *ratio* stays constant forever. In pure compounding, a head start is permanent.

Page: https://tryals.app/practice/business-i/differential-equations/two-deposits-obey-the-same-ode-dotx-0-05x-but-start-at-different

### 6. Order the phases of a logistic product adoption.

**Answer:**

1. Launch: few users, growth slow in absolute terms
2. Near-exponential takeoff, word of mouth compounds
3. Inflection at half the market: fastest growth
4. Deceleration as untapped customers thin out
5. Plateau at capacity $K$

**Why:** Slow, explosive, peaking at $K/2$, braking, saturated, the S-curve read left to right. Knowing *where on the S you are* is worth more than the whole forecast.

Page: https://tryals.app/practice/business-i/differential-equations/order-the-phases-of-a-logistic-product-adoption

### 7. Order the steps of ODE-based forecasting.

**Answer:**

1. State the rate law the business obeys
2. Solve the ODE for the general trajectory
3. Fit $x_0$ and $r$ to observed data
4. Project the fitted curve forward
5. Re-check the model when reality bends away

**Why:** Law → solution → calibration → projection → revision. Step 5 is where exponential fits get demoted to logistic ones, usually just after the optimistic board presentation.

Page: https://tryals.app/practice/business-i/differential-equations/order-the-steps-of-ode-based-forecasting

### 8. A startup projects 20%/month growth "indefinitely" in its pitch deck. Which mathematical objection is the sharpest?

A. 20% is not a legal growth rate
B. The ODE $\dot{x} = 0.2x$ has no solution
C. Exponentials outgrow any finite market
D. Growth rates must decline linearly by definition

**Answer:** C. Exponentials outgrow any finite market

**Why:** At 20%/month the user base multiplies ~9× yearly; in a few years it exceeds humanity. The exponential is locally right and globally impossible, every honest forecast eventually swaps in the $(1 - x/K)$ brake.

Page: https://tryals.app/practice/business-i/differential-equations/a-startup-projects-20-month-growth-indefinitely-in-its-pitch-deck