Definition, Order and Degree of a Differential Equation

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter introduces differential equations — equations involving an unknown function and its derivatives — building on the integration toolkit from the previous two chapters to actually SOLVE them.

What is a differential equation?

A differential equation is an equation containing an unknown function $y=y(x)$ together with one or more of its derivatives, e.g. $\frac{dy}{dx}=2x$ or $\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=0$.

Order and degree

The order of a differential equation is the order of the HIGHEST derivative that appears in it. The degree is the power of that highest-order derivative, once the equation has been written as a polynomial in its derivatives (no radicals or fractional powers of derivatives).

Worked reasoning

For $\frac{d^2y}{dx^2}+3\left(\frac{dy}{dx}\right)^2+5y=0$: the highest derivative is $\frac{d^2y}{dx^2}$ (second order), and since the equation is already a polynomial in its derivatives with $\frac{d^2y}{dx^2}$ appearing to the power $1$, the order is 2 and the degree is 1.

For $\left(\frac{d^2y}{dx^2}\right)^3+\left(\frac{dy}{dx}\right)^2=x$: the highest derivative is again $\frac{d^2y}{dx^2}$ (order 2), but here it is raised to the power $3$, so the degree is 3.