# First-Order Differential Equations

Business I · Mathematics II · https://tryals.app/learn/business-i/first-order-differential-equations

## First-Order ODEs & Methods

A **differential equation** relates a function to its derivatives, describing _how_ something changes. **First-order ODEs** involve only the first derivative: $\frac{dy}{dt} = f(t, y)$.

**Separable equations** occur when $f(t,y) = g(t) h(y)$. Separate variables and integrate: 

$$\int \frac{dy}{h(y)} = \int g(t)\, dt$$

**Linear first-order ODEs** follow $\frac{dy}{dt} + p(t) y = q(t)$. Solve using the **integrating factor** $\mu(t) = e^{\int p(t)\, dt}$:

$$y(t) = \frac{1}{\mu(t)} \int \mu(t) q(t)\, dt$$

| Type | Form | Method |
| --- | --- | --- |
| Separable | $dy/dt = g(t)h(y)$ | Separate and integrate |
| Linear | $dy/dt + p(t)y = q(t)$ | Integrating factor $e^{\int p\,dt}$ |
| Exponential | $dy/dt = ky$ | Direct: $y_0 e^{kt}$ |

> **Common pitfall:** Dividing by $h(y)$ assumes $h(y) \neq 0$. Always check constant solutions where $h(y) = 0$ separately.

## Economic Models & Stability

The **exponential growth/decay model** $\frac{dy}{dt} = ky$ yields $y(t) = y_0 e^{kt}$. If $k > 0$, it is growth; if $k < 0$, it is decay.

Key economic applications include:
- **Compound interest**: $\frac{dA}{dt} = rA \Rightarrow A_0 e^{rt}$
- **Price adjustment**: $\frac{dp}{dt} = \alpha(D - S)$
- **Solow growth**: $\dot{k} = sf(k) - \delta k$
- **Advertising decay**: $A(t) = A_0 e^{-\delta t}$

The steady state occurs where $dy/dt = 0$. For $\dot{y} = f(y)$ near equilibrium $y^*$, stability depends on the derivative sign.

| Derivative | Stability | Behavior |
| --- | --- | --- |
| $f'(y^*) < 0$ | **Stable** | Deviations shrink |
| $f'(y^*) > 0$ | **Unstable** | Deviations grow |

## Practice questions

8 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Sort these into growth models vs decay models.

**Answer:**

- Growth ($k > 0$): Continuous compound interest at 5%, Population with positive birth rate, Viral spread in early phase
- Decay ($k < 0$): Radioactive decay, Brand awareness without advertising, Asset depreciation

**Why:** Interest, population, and viral spread grow; decay, awareness loss, and depreciation shrink. The sign of $k$ tells the story.

Page: https://tryals.app/practice/business-i/first-order-differential-equations/sort-these-into-growth-models-vs-decay-models

### 2. $y = y_0 e^{kt}$ solves $\dot{y} = ky$ because the exponential is its own derivative up to a constant.

**Answer:** True

**Why:** True. $\frac{d}{dt}(y_0e^{kt}) = k y_0 e^{kt} = ky$, which is exactly what the equation demands. Any quantity whose rate of change is proportional to its size — compound interest, unchecked population, radioactive decay — follows this form, with the sign of $k$ deciding growth or decay.

Page: https://tryals.app/practice/business-i/first-order-differential-equations/y-y0-e-kt-solves-doty-ky-because-the-exponential-is-its-own

### 3. At 7% continuous growth, the Rule of 70 says doubling time is roughly $70/7 = 10$ years. **Estimate $e^{0.7}$.**

**Answer:** 2.014 (within ±0.15)

**Why:** $e^{0.7} \approx 2.014$, close to 2, confirming the Rule of 70 approximation. At 7% continuous growth, you roughly double in 10 years.

Page: https://tryals.app/practice/business-i/first-order-differential-equations/at-7-continuous-growth-the-rule-of-70-says-doubling-time-is-roughly

### 4. An exponential decay model stays accurate all the way down to the last few units of the quantity.

**Answer:** False

**Why:** False. Exponential decay is a continuous average over many units; once few remain, individual randomness dominates and the smooth curve stops describing anything. The model never reaches zero mathematically, and the real quantity does, the mismatch is a sign of leaving the model's domain.

Page: https://tryals.app/practice/business-i/first-order-differential-equations/an-exponential-decay-model-stays-accurate-all-the-way-down-to-the

### 5. Which spreadsheet formula computes continuous compound interest $A = 1000 \times e^{0.05 \times 10}$?

A. =1000*(1+0.05)^10
B. =1000*EXP(0.05*10)
C. =1000*0.05^10
D. =1000*LN(0.05*10)

**Answer:** B. =1000*EXP(0.05*10)

**Why:** EXP() computes $e^x$. The formula =1000*EXP(0.05*10) = =1000*EXP(0.5) gives continuous compounding. Option C gives discrete compounding; option D uses the wrong function.

Page: https://tryals.app/practice/business-i/first-order-differential-equations/which-spreadsheet-formula-computes-continuous-compound-interest-a

### 6. Half-life from a decay rate $k$ is computed as $\ln 2 / |k|$ rather than $2/k$. Why does a logarithm appear?

A. Because halving is reached after a fixed multiple of $1/k$
B. Because $\ln 2$ is an approximation to the exact factor 2
C. Because the logarithm converts the rate into the correct units
D. Because half-life is defined logarithmically by convention

**Answer:** A. Because halving is reached after a fixed multiple of $1/k$

**Why:** The logarithm is what inverts the exponential. It is also why half-life is constant: the time to halve does not depend on where you start, which is peculiar to exponential decay.

Page: https://tryals.app/practice/business-i/first-order-differential-equations/half-life-from-a-decay-rate-k-is-computed-as-ln-2-k-rather-than

### 7. Match each ODE to its economic model.

**Answer:**

- $\dot{A} = rA$ → Continuous compound interest
- $\dot{p} = \alpha(D - S)$ → Price adjustment toward equilibrium
- $\dot{k} = sf(k) - \delta k$ → Solow capital accumulation
- $\dot{A} = -\delta A$ → Advertising awareness decay

**Why:** Each economic process has a differential equation: growth, equilibrium seeking, accumulation, and decay.

Page: https://tryals.app/practice/business-i/first-order-differential-equations/match-each-ode-to-its-economic-model

### 8. A separable differential equation and a non-separable one may describe equally simple business situations. What does separability actually reflect?

A. That the solution can be written in closed form, which non-separable equations never allow
B. That the underlying process is linear
C. That the equation has a unique solution, which non-separable equations may lack
D. That time and level can be factored apart in the model

**Answer:** D. That time and level can be factored apart in the model

**Why:** Separability is a convenience of form, not a claim about the dynamics, the logistic equation is separable and produces the whole S-curve. Option 4 overstates it too: many non-separable equations have closed forms, integrating factors being the standard route.

Page: https://tryals.app/practice/business-i/first-order-differential-equations/a-separable-differential-equation-and-a-non-separable-one-may