Let m and n be the order and the degree respectively of the differential equation \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^3 + 4\left(\displaystyle\frac{dy}{dx}\right)^4 - xy = 0\). Then
(A) \(m = 2\)
(B) \(n = 4\)
(C) \(m + n = 5\)
(D) \(mn = 8\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
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 is \(\frac{d^2y}{dx^2}\), so
\(m = 2\)
It appears with power 3 (the power 4 belongs to the lower derivative \(\frac{dy}{dx}\)), so
\(n = 3\)
So m = 2 and n = 3, giving m + n = 5 and mn = 6.
Checking the statements
(A) \(m = 2\) — Correct. The highest derivative is the second.
(B) \(n = 4\) — Incorrect. The degree is 3, the power of \(\frac{d^2y}{dx^2}\).
(C) \(m + n = 5\) — Correct. \(2 + 3 = 5\).
(D) \(mn = 8\) — Incorrect. \(mn = 2 \times 3 = 6\).