Differential Equation: Definition, Order and Degree

A differential equation is an equation that connects an independent variable (say $x$), a dependent variable (say $y$) and one or more derivatives of $y$ with respect to $x$. Because a derivative measures a rate of change, a differential equation is the natural language for any situation described by “the rate at which something changes” — the growth of a population, the depreciation of a machine, the cooling of a body, or the accumulation of interest.

Some examples:

Order of a differential equation is the order of the highest derivative appearing in it.

Degree of a differential equation is the power (exponent) of that highest-order derivative, after the equation has been made free of radicals and fractional powers of the derivatives and written as a polynomial in the derivatives. If the equation cannot be expressed as such a polynomial (for instance when a derivative appears inside a trigonometric, logarithmic or exponential function), the degree is not defined.

Illustrations.

• $\dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} + 2y = 0$: highest derivative is $\dfrac{d^2y}{dx^2}$ (order 2); its power is $1$ (degree 1).
• $\left(\dfrac{dy}{dx}\right)^3 + y = x$: highest derivative $\dfrac{dy}{dx}$ (order 1); its power is $3$ (degree 3).
• $\sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} = \dfrac{d^2y}{dx^2}$: square both sides first → $1 + \left(\dfrac{dy}{dx}\right)^2 = \left(\dfrac{d^2y}{dx^2}\right)^2$. Now it is a polynomial in the derivatives: order 2, degree 2.

Maharashtra's Std XII Commerce Mathematics and Statistics syllabus develops differential equations from the same standard principles of calculus taught across Indian higher-secondary and commerce-mathematics curricula — order and degree, formation, the variables-separable / homogeneous / linear solution methods, and applications to growth and decay, all covered in this chapter.