Order and Degree of a Differential Equation

Two simple numbers describe the “shape” of a differential equation before any attempt is made to solve it: its order and its degree.

The order of a differential equation is the order of the highest derivative that appears in it. If only $\dfrac{dy}{dx}$ appears, the order is $1$; if $\dfrac{d^2y}{dx^2}$ appears (even alongside $\dfrac{dy}{dx}$), the order is $2$; and so on. Order is read off directly — no algebraic manipulation is needed to find it.

The degree is the power (exponent) to which the highest-order derivative is raised, once the equation has been written as a polynomial in the derivatives — that is, after clearing any radicals or fractional/negative powers that involve a derivative. Degree, unlike order, sometimes requires a preliminary algebraic step before it can be read off.

Illustration — degree read off directly: in $\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 = x$, the highest derivative is $\dfrac{d^2y}{dx^2}$ (so the order is $2$), and it already appears raised to a whole-number power with no radical involved, so the degree is read directly as $3$.

Illustration — degree found only after clearing a radical: in $\sqrt{1+\left(\dfrac{dy}{dx}\right)^2} = x\dfrac{d^2y}{dx^2}$, the highest derivative $\dfrac{d^2y}{dx^2}$ is not yet free of radicals elsewhere in the equation, so the degree cannot be read off in this form. Squaring both sides first gives $1+\left(\dfrac{dy}{dx}\right)^2 = x^2\left(\dfrac{d^2y}{dx^2}\right)^2$ — now every derivative appears as a whole-number power, and the highest-order derivative $\dfrac{d^2y}{dx^2}$ is raised to the power $2$, so the degree is $2$ (the order remains $2$ throughout, since squaring did not change which derivative is of highest order).