Definition of a Differential Equation

An ordinary differential equation is an equation that connects an unknown function $y = y(x)$ of a single independent variable $x$ with one or more of its derivatives — $\dfrac{dy}{dx}$, $\dfrac{d^2y}{dx^2}$, and so on. For example, $\dfrac{dy}{dx} = 2x$, or $\dfrac{dy}{dx} + y = x^2$, or $\dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} = 0$ are all differential equations. This chapter stays entirely within ordinary differential equations of one variable — the equations already met in the Limits and Derivatives (Class XI) and Integration chapters simply reappear here, now read backwards: instead of being given $y$ and asked to find its derivative, we are given a relationship involving the derivative and asked to find $y$ itself.

The crucial difference from an ordinary algebraic equation like $2x+3=7$ is what counts as a “solution.” An algebraic equation is satisfied by a finite list of numbers (here, just $x=2$). A differential equation, by contrast, is satisfied by a function — an entire rule $y = f(x)$ that makes the equation true for every value of $x$ in some domain, not just one number.

Illustration: consider $\dfrac{dy}{dx} = 2x$. Any function of the form $y = x^2 + c$, for any constant $c$, satisfies this equation, because $\dfrac{d}{dx}(x^2+c) = 2x$ regardless of what $c$ is. This whole family of functions, written with the arbitrary constant still present, is called the general solution of the differential equation. If an extra condition is supplied — say, we are told $y=5$ when $x=0$ — substituting into $y=x^2+c$ gives $5 = 0+c$, so $c=5$; the single function $y=x^2+5$ picked out this way is called a particular solution.

WBCHSE's Business Mathematics and Basic Statistics syllabus draws on the same differential-equations principles — order, degree, formation from a family of curves, and the variable-separable solving technique — that are taught across Indian commerce-mathematics and applied-calculus curricula, scoped here specifically to simple, algebraic cases suited to a business-mathematics course rather than a full calculus course.