The sum of the order and the degree of the differential equation \(\left(\displaystyle\frac{d^3y}{dx^3}\right)^2 + \left(\displaystyle\frac{d^2y}{dx^2}\right)^5 + \displaystyle\frac{dy}{dx} + y = 0\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5
The order is the order of the highest derivative present. The degree is the power of that highest derivative, when the equation is a polynomial in the derivatives.
The highest derivative here is the third derivative, so
\(\text{order} = 3\)
The equation is a polynomial in the derivatives, and \(\frac{d^3y}{dx^3}\) appears with power 2, so
\(\text{degree} = 2\)
Adding:
\(3 + 2 = 5\)
Why the other options are incorrect
Option (1) 7 — Incorrect. This is 2 + 5: it takes the second derivative and its power 5, but the third derivative is the highest one present.
Option (2) 8 — Incorrect. This is 3 + 5: the power 5 belongs to the lower derivative \(\frac{d^2y}{dx^2}\), not to the highest one.
Option (4) 6 — Incorrect. This is the product \(3 \times 2\), not the sum.