1 — What Is an Ordinary Differential Equation?
Definition
An ordinary differential equation (ODE) is an equation that relates an unknown function $y(t)$ (or $y(x)$) to one or more of its derivatives. For example:
$$ \frac{dy}{dt} = 3y $$Here the unknown is not a number $y$; it is a function $y(t)$ whose derivative equals three times itself.
Ordinary vs. Partial — One Honest Sentence
An ODE involves derivatives with respect to a single independent variable; a partial differential equation (PDE) involves partial derivatives with respect to multiple independent variables (e.g., both $x$ and $t$). In this course we study only ODEs.
Order
The order of an ODE is the highest derivative that appears in the equation. Read it off directly:
| Equation | Highest Derivative | Order |
|---|---|---|
| $y' = 2y$ | $y'$ | 1 |
| $y'' + 4y = 0$ | $y''$ | 2 |
| $y''' - y' + y = e^t$ | $y'''$ | 3 |
| $y^{(4)} + y'' = \sin t$ | $y^{(4)}$ | 4 |
2 — Where the Equations Come From: The Modeling Craft
Every ODE in this course began as a sentence about the real world. The first skill of the subject is translation: turning a verbal statement into a differential equation. The two key moves are:
- Rates become derivatives: “the rate of change of $P$” becomes $\dfrac{dP}{dt}$.
- Proportionality becomes a coefficient: “proportional to $P$” becomes $kP$ for some constant $k$.