dependent and independent variables

Every differential equation has two kinds of letters in it, playing two very different roles, and getting them straight prevents a great deal of confusion. One letter is the input you are free to choose or that simply ticks forward on its own — like time. The other is the output, the quantity whose behaviour you are trying to discover — like temperature, population, or position. Knowing which is which is the first step to reading any equation.

Precisely, the independent variable is the one we differentiate WITH RESPECT TO — typically x or t — and we think of it as ranging freely along an axis. The dependent variable is the unknown quantity that responds, written as a function of the independent one, like y(x) or P(t); it 'depends on' the independent variable. In dy/dt = k y, time t is independent (it just advances) and y is dependent (it is whatever responds to the rule). An ordinary differential equation has exactly one independent variable; a partial differential equation has two or more.

Identifying the variables correctly tells you the geometry of the problem — what you plot on which axis (independent horizontal, dependent vertical) — and it is what distinguishes an ordinary from a partial equation. It also clears up which symbols are 'allowed' to make an equation nonlinear: only the dependent variable and its derivatives do that, never the independent one. A common beginner's error is to treat a coefficient like t or x as if it were the unknown; it is not — it is just the stage on which the unknown plays out.