partial differential equation
/ PDE (pee-dee-ee) /
Now imagine the thing you track depends on MORE THAN ONE varying quantity at once. The temperature inside a metal bar depends both on where you are along the bar and on what time it is. The height of a ripple on a pond depends on position across the surface and on time. To describe how such a quantity changes, you need to talk about its rate of change in each direction separately — and that is what a partial derivative does.
Precisely, a partial differential equation relates an unknown function of SEVERAL independent variables to its partial derivatives — derivatives taken with respect to one variable while holding the others fixed, written with the curly symbol, as in du/dt and d^2u/dx^2 (partial). The heat equation du/dt = c d^2u/dx^2 (partial) is a classic example: it relates how temperature u(x, t) changes in time to how it curves in space. Because several directions of change interact, partial differential equations are generally much harder than ordinary ones.
We mention partial differential equations here only to draw the boundary of our subject: this course is about ORDINARY differential equations, with a single independent variable. Partial differential equations are their own large field, but the two are deeply linked — a standard technique called separation of variables turns certain partial differential equations INTO a set of ordinary ones, which is one reason mastering the ordinary case first is so valuable.
The wave equation d^2u/dt^2 = c^2 d^2u/dx^2 (all partial) is a PDE: the unknown u(x, t) depends on two variables, position x and time t, and partial derivatives in both appear. No single-variable ODE can capture this.
Two or more independent variables, partial derivatives — beyond this course, but worth recognizing.
The dividing line is the NUMBER of independent variables, not how complicated the equation looks. One independent variable means ordinary; two or more means partial.