Q.Find the order and degree of $\dfrac{d^2y}{dx^2}+3\left(\dfrac{dy}{dx}\right)^2+5y=0$.

A differential equation is any equation that contains at least one derivative — ordinary or partial — of an unknown function. If only ordinary derivatives of a function of a single independent variable appear, it is an Ordinary Differential Equation (ODE); if partial derivatives of a function of two or more independent variables appear, it is a Partial Differential Equation (PDE). This chapter deals only with ODEs.

Order. The order of a differential equation is the order of the highest derivative that appears in it. If the highest derivative of $y$ present is the $k$th derivative, the order is $k$ (a positive integer). For example, $\dfrac{d^3y}{dx^3}-5\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}=0$ has order $3$.

Degree. The degree is defined only once the equation has been written in polynomial form in its derivatives — every derivative free of fractional powers or roots, and the highest-order derivative not sitting inside a transcendental function (sine, log, exponential, …) or having a coefficient that is itself transcendental in the derivatives. Once in that form, the degree is the integral power to which the highest-order derivative is raised.

Working method.

1. If radicals or fractional powers appear on a derivative, isolate that term and raise both sides to the appropriate power to clear it (square, cube, …) — this can raise or lower the apparent order/degree, so always simplify to the true polynomial form first.
2. If the equation contains an integral of $y$ (not a derivative), differentiate the whole equation once more with respect to $x$ to eliminate the integral sign before reading off order and degree.
3. If, even after full simplification, the highest-order derivative sits inside a sine/cosine/log/exponential — or a lower-order derivative does, since that also breaks the "polynomial in the derivatives" requirement — the equation cannot be written in polynomial form, and its degree is not defined (the order can still be stated).

The degree, whenever it exists, is always a positive integer.

Solutions and constants. A solution is an expression for $y$ in terms of $x$ (or vice versa) that satisfies the equation; it need not exist, and need not be unique. The general solution carries as many independent arbitrary constants as the order of the equation; assigning particular numerical values to those constants (usually via extra given conditions) gives a particular solution, which has zero arbitrary constants. Correspondingly, eliminating $n$ arbitrary constants from a family of curves always produces a differential equation of order $n$ — eliminating one constant gives a first-order equation, eliminating two gives a second-order equation, and so on.