Q.The order and degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^3-3\frac{d^3y}{dx^3}+2\left(\frac{dy}{dx}\right)^4=y^4$ are:
(A) 1, 4
(B) 3, 4
(C) 2, 4
(D) 3, 2

Order: the highest derivative present is $\frac{d^3y}{dx^3}$, so the order is $3$.

Degree: the equation is already polynomial in its derivatives, and the highest-order derivative $\frac{d^3y}{dx^3}$ appears to the first power (in the term $-3\frac{d^3y}{dx^3}$), so the degree is $1$. …

The equation is

Order

Scan for the highest derivative. The derivatives present are $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$ and $\frac{d^3y}{dx^3}$; the highest is the third derivative. So the order is 3.

Degree …

Method: Determine order and degree of a differential equation

Use this classification before choosing any solving technique.

Steps

Step 1: Order = the highest derivative that appears.

Scan every term and pick the most-differentiated one. A third derivative present anywhere makes the order $3$, regardless of the powers on lower derivatives.

Step 2: Check the equation is polynomial in the derivatives.

There must be no radicals, no fractional powers, and no derivative trapped inside a transcendental function. Only then is the degree defined. …

Common Mistakes

Mistake 1: Taking the degree from a lower-order derivative's power.

Why it's wrong: the cube on $\frac{d^2y}{dx^2}$ and the fourth power on $\frac{dy}{dx}$ tempt answers of degree $4$ (options A–C). Degree is the power of the highest-order derivative, which is $\frac{d^3y}{dx^3}$, appearing here to the first power. Correct approach: order $=3$, degree $=1$. …