Order of a Differential Equation

Order of a Differential Equation

The order of a differential equation is the order of the highest-order derivative of the dependent variable that appears in the equation. This is the first and most fundamental classification.

Consider the following three differential equations:

1. $\frac{dy}{dx} = e^{x}$ ... (6)
2. $\frac{d^{2}y}{dx^{2}} + y = 0$ ... (7)
3. $\left(\frac{d^{3}y}{dx^{3}}\right)^{2} + x\left(\frac{d^{2}y}{dx^{2}}\right)^{3} = 0$ ... (8)

The highest derivative present is $\frac{dy}{dx}$ in (6), $\frac{d^{2}y}{dx^{2}}$ in (7), and $\frac{d^{3}y}{dx^{3}}$ in (8). So their orders are 1, 2, and 3 respectively. …