Definition of a Differential Equation

An equation that involves an independent variable, a dependent variable, and one or more derivatives of the dependent variable with respect to the independent variable is called a differential equation. For example,

are all differential equations: in each, $y$ is the dependent variable, $x$ is the independent variable, and the equation relates $y$ (and possibly $x$) to derivatives of $y$.

Ordinary vs. partial. When the dependent variable depends on only one independent variable, so that every derivative appearing is an ordinary derivative such as $\dfrac{dy}{dx}$ or $\dfrac{d^2y}{dx^2}$, the equation is called an ordinary differential equation (ODE). When the dependent variable depends on two or more independent variables and the equation involves partial derivatives (e.g. $\dfrac{\partial z}{\partial x}$), it is called a partial differential equation (PDE). This chapter, in line with the WBCHSE syllabus, deals only with ordinary differential equations, and "differential equation" below always means "ordinary differential equation."

Notation. Besides $\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}, \dfrac{d^3y}{dx^3},\ldots$, the shorthand $y', y'', y''', \ldots$ (or $y_1, y_2, y_3,\ldots$) is often used for the first, second, third, $\ldots$ derivatives of $y$ with respect to $x$.

Where differential equations arise. Many real quantities are easier to describe through how fast they change than through an explicit formula: the rate at which a population grows is often taken proportional to the population already present ($\frac{dP}{dt} = kP$); the rate at which a hot object cools is taken proportional to the difference between its temperature and the surrounding temperature (Newton's law of cooling); the slope of a curve at each point may be prescribed instead of the curve's equation itself. Each such rate-relationship, once written down, is a differential equation, and solving it (Sections 3–7) recovers the quantity itself as an explicit function.