The product of the order and degree of the differential equation \( \left(\displaystyle\frac{d^3y}{dx^3}\right)^2-3\displaystyle\frac{d^2y}{dx^2}+2\left(\displaystyle\frac{dy}{dx}\right)^4=y^4 \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 6
The order is the order of the highest derivative present, and the degree is the power of that highest derivative when the equation is a polynomial in the derivatives.
The highest derivative is \( \frac{d^3y}{dx^3} \), so the order is 3.
It occurs with power 2, so the degree is 2.
\( \displaystyle \text{order}\times\text{degree}=3\times 2=6 \)
Why the other options are incorrect
Option (1) 12 — Incorrect. 12 = 3 × 4 takes the degree as 4, the power of \( \frac{dy}{dx} \); the degree is the power of the highest derivative, not the highest power.
Option (3) 4 — Incorrect. 4 = 2 × 2 takes the order as 2, but the equation contains \( \frac{d^3y}{dx^3} \).
Option (4) 8 — Incorrect. 8 = 2 × 4 makes both mistakes: order 2 and degree 4.