The product of order and degree of the differential equation
\( x\displaystyle\frac{d^3y}{dx^3}=\left(1+\left(\frac{dy}{dx}\right)^2\right)^4 \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
The highest derivative present is \( \frac{d^3y}{dx^3} \), so the order is 3.
The equation is a polynomial in the derivatives, and the highest-order derivative \( \frac{d^3y}{dx^3} \) appears only to the power 1, so the degree is 1.
\( \displaystyle \text{order}\times\text{degree}=3\times 1=3 \)
Why the other options are incorrect
Option (1) 12 — Incorrect. 12 = 3 × 4 takes the outer power 4 as the degree, but that power belongs to \( \frac{dy}{dx} \), not to the highest derivative.
Option (3) 8 — Incorrect. 8 is the highest power of \( \frac{dy}{dx} \) after expanding the right side, not the order or the degree.
Option (4) 6 — Incorrect. 6 = 3 × 2 takes the power 2 of \( \frac{dy}{dx} \) as the degree.