Q.The degree of the differential equation $\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^3+6y^5=0$ is:
(A) 1
(B) 2
(C) 3
(D) 5

Concept: Order of a differential equation is the highest derivative present; degree is the power of the highest derivative after the equation is made polynomial in derivatives (no radicals, no fractions of derivatives).

Step 1: Identify the highest derivative. Here it is $\frac{d^2y}{dx^2}$ — so the order is 2. …

The degree of a differential equation is defined only when the equation is a polynomial in the derivatives. It is the power of the highest-order derivative present, after the equation has been made free of radicals and fractions in the derivatives.

Here, the highest derivative is $\frac{d^2y}{dx^2}$ (order 2). The equation is:

This is already a polynomial in $\frac{d^2y}{dx^2}$ and $\frac{dy}{dx}$ — no square roots, no fractional powers of derivatives. The highest-order derivative $\frac{d^2y}{dx^2}$ appears with exponent 1. So the degree is simply 1.

Method: Degree from the Highest-Order Derivative Only

Degree is the power of the highest-order derivative once the equation is polynomial in its derivatives. Powers of lower derivatives or of $y$ itself are irrelevant.

Steps

Step 1: Identify the highest-order derivative.

In $\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^3+6y^5=0$, the highest is $\frac{d^2y}{dx^2}$ (order $2$).

Step 2: Check the equation is polynomial in derivatives.

There are no radicals or transcendental wrappers on any derivative, so a degree exists. …

Common Mistakes

Mistake 1: Reading the degree as $3$ from $\left(\frac{dy}{dx}\right)^3$.

Why it's wrong: degree is the power of the highest-order derivative, which is $\frac{d^2y}{dx^2}$ (power $1$), not the cube on the lower-order $\frac{dy}{dx}$. Correct approach: locate the highest-order derivative first, then read its power. …