Q.Determine the order and degree (if exists) of the differential equation $x^2\dfrac{d^2y}{dx^2}+\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{1/2}=0$

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. …