The product of the order and the degree of the differential equation \(\displaystyle\frac{d^2y}{dx^2}\left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{dy}{dx}\right)^4 = y\) 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 (the equation is a polynomial in its derivatives).
The highest derivative is \(\frac{d^3y}{dx^3}\), so
\( \text{order} = 3 \)
It appears with power 2 (the factor \(\frac{d^2y}{dx^2}\) in front does not change this power), so
\( \text{degree} = 2 \)
The product is
\( 3 \times 2 = 6 \)
Why the other options are incorrect
Option (1) 12 — Incorrect. This uses the power 4 of \(\frac{dy}{dx}\); the degree is the power of the highest derivative only.
Option (3) 8 — Incorrect. This multiplies the order 2 of the second derivative by the power 4 of \(\frac{dy}{dx}\); neither is the order or degree here.
Option (4) 3 — Incorrect. This takes the degree as 1; the third derivative is squared.