Basic Concepts

9.2 Basic Concepts

You already know equations like $x^2 - 3x + 3 = 0$ and $x + y = 7$, which involve only the variables themselves. Now consider

This contains the derivative $\frac{dy}{dx}$ in addition to $x$ and $y$. In general, an equation that involves one or more derivatives of a dependent variable with respect to one or more independent variables is called a differential equation.

Ordinary vs. Partial Differential Equations

• If all derivatives are with respect to only one independent variable, the equation is an ordinary differential equation.
• If derivatives are with respect to more than one independent variable, it is a partial differential equation.

For example,

is ordinary — $y$ depends only on $x$, and all derivatives are taken with respect to $x$.

Notation for Derivatives

Writing $\frac{dy}{dx}$, $\frac{d^2 y}{dx^2}$, etc. repeatedly is cumbersome, so we use the compact notations:

• $y' = \frac{dy}{dx}$ (first derivative) …