Order and Degree

Two numbers are attached to every differential equation before it is solved: its order and its degree.

Order. The order of a differential equation is the order of the highest-order derivative appearing in it. If the highest derivative present is $\dfrac{d^2y}{dx^2}$, the equation has order $2$; if it is $\dfrac{dy}{dx}$, order $1$; and so on. Order does not depend on any power to which that derivative is raised -- only on how many times $y$ has been differentiated.

Degree. The degree of a differential equation is the power of the highest-order derivative, provided the equation is a polynomial equation in its derivatives -- i.e. provided every derivative appears only raised to a non-negative integer power, with no derivative inside a radical, a fractional power, a trigonometric function, an exponential, a logarithm, or a denominator. If radicals or fractional powers involving derivatives are present, the equation must first be rearranged (e.g. by squaring, cubing, or clearing denominators) into a form free of them before the degree can be read off.

Worked illustration. For $\left(\dfrac{d^3y}{dx^3}\right)^{2/3} = \dfrac{dy}{dx} + y$, the highest derivative $\dfrac{d^3y}{dx^3}$ appears with the fractional power $2/3$, so the degree cannot yet be read off. Cubing both sides clears the fractional power:

Now the equation is polynomial in the derivatives: the highest derivative is $\dfrac{d^3y}{dx^3}$ (order $3$), raised to the power $2$ (degree $2$). …