Which of the following differential equations has degree 2?

The correct answer is Option (2) → \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 = x\)

The degree is the highest power of the highest derivative, and it is defined only when the equation is a polynomial in the derivatives.

In Option (2) the highest derivative is \(\frac{d^2y}{dx^2}\), and it appears squared. The power 3 belongs to the lower derivative \(\frac{dy}{dx}\). So

    \(\text{degree} = 2\)

Why the other options are incorrect

• Option (1) \(\displaystyle\frac{d^3y}{dx^3} + \left(\frac{d^2y}{dx^2}\right)^2 = 0\) — Incorrect. The highest derivative is \(\frac{d^3y}{dx^3}\), with power 1, so the degree is 1.

• Option (3) \(\displaystyle\frac{d^2y}{dx^2} + y^2 = \sin x\) — Incorrect. \(\frac{d^2y}{dx^2}\) has power 1; the square is on y, not on a derivative. The degree is 1.

• Option (4) \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^2 + \sin\left(\frac{dy}{dx}\right) = 0\) — Incorrect. \(\sin\left(\frac{dy}{dx}\right)\) is not a polynomial in \(\frac{dy}{dx}\), so the degree is not defined.