# First-Order Differential Equations

Physics I · Differential Equations and Vector Calculus · https://tryals.app/learn/physics-i/first-order-differential-equations

## First-Order ODEs

A **first-order differential equation** involves only first derivatives and has the general form:

$$\frac{dy}{dx} = f(x, y)$$

**Separable equations** occur when $f(x,y) = g(x)h(y)$. Rearrange and integrate both sides:

$$\int \frac{dy}{h(y)} = \int g(x)\,dx$$

**Linear first-order equations** match the standard form $y' + P(x)y = Q(x)$. Solve them using the **integrating factor** $\mu$:

$$\mu = e^{\int P\,dx}, \quad y = \frac{1}{\mu}\int \mu\,Q\,dx$$

**Worked Example**: Radioactive decay is modeled by $\frac{dN}{dt} = -\lambda N$. Separating variables yields the solution $N = N_0 e^{-\lambda t}$, showing exponential decrease over time.

## Core Types & Pitfalls

First-order equations fall into four main categories, each requiring a specific solution strategy:

| Type | Standard Form | Solution Method |
| --- | --- | --- |
| **Separable** | $y' = g(x)h(y)$ | Separate variables |
| **Linear** | $y' + Py = Q$ | Integrating factor |
| **Exact** | $M\,dx + N\,dy = 0$ | Potential function |
| **Bernoulli** | $y' + Py = Qy^n$ | Substitution $v = y^{1-n}$ |

**Physics Link**: Newton's law of cooling, $\frac{dT}{dt} = -k(T - T_{\text{env}})$, is a linear first-order ODE governing thermal equilibrium.

**Common Pitfall**: The general solution of a first-order ODE requires an arbitrary constant. Forgetting **+C** (or ignoring initial conditions) yields **a** specific solution rather than the complete solution required for your problem.

## Practice questions

4 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A first-order ODE is one whose unknown function appears raised only to the first power.

**Answer:** False

**Why:** False. **Order** is the highest *derivative* present; the power the unknown is raised to is its **degree**. $(y')^{2} = y$ is first order and second degree, while $y'' = y$ is second order and first degree. Conflating the two matters because linearity, which decides whether superposition applies, depends on degree, not order.

Page: https://tryals.app/practice/physics-i/first-order-differential-equations/a-first-order-ode-is-one-whose-unknown-function-appears-raised-only

### 2. Some equations can be solved by separation AND by an integrating factor. Why does having two valid methods not make the choice arbitrary?

A. Because one route is usually far shorter for the equation
B. Because the two methods give different solutions that must be reconciled
C. Because integrating factors work only for nonlinear equations
D. Because only one method is rigorous and the other is a shortcut

**Answer:** A. Because one route is usually far shorter for the equation

**Why:** Both are valid, and one is quicker. Reading the equation for its structure before committing is what makes method selection a skill rather than a lookup.

Page: https://tryals.app/practice/physics-i/first-order-differential-equations/some-equations-can-be-solved-by-separation-and-by-an-integrating

### 3. Match each real-world statement to the differential equation that models it.

**Answer:**

- Population grows in proportion to its size → $y' = ky$
- Coffee cools toward room temperature $T_r$ → $y' = -k(y - T_r)$
- A falling body approaches terminal velocity → $v' = g - cv$
- Growth slows as capacity $M$ is approached → $y' = ky(1 - y/M)$

**Why:** Modelling is translation: each phrase about rates becomes a term in $y'$. Notice how "approaches" always produces a stabilizing term that vanishes at equilibrium, the mathematical signature of a system settling down.

Page: https://tryals.app/practice/physics-i/first-order-differential-equations/match-each-real-world-statement-to-the-differential-equation-that

### 4. A first-order ODE has a family of solutions and an initial condition picks one out. Why does the physics require the extra condition rather than the equation alone?

A. Because the equation states a LAW and the condition picks a copy
B. Because integration introduces an error that the condition corrects
C. Because first-order equations have infinitely many laws to choose between
D. Because the equation is underdetermined and therefore incomplete as physics

**Answer:** A. Because the equation states a LAW and the condition picks a copy

**Why:** Law and state are separate inputs. Newton’s second law does not tell you where the planet is today, the constant of integration is exactly the room left for that.

Page: https://tryals.app/practice/physics-i/first-order-differential-equations/a-first-order-ode-has-a-family-of-solutions-and-an-initial-condition