Let m and n be respectively the order and the degree of the differential equation \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^2 + x\left(\displaystyle\frac{dy}{dx}\right)^3 = 4y\).
(A) \(m = 2\)
(B) \(n = 3\)
(C) The general solution of this differential equation contains two arbitrary constants.
(D) \(m = n\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
The highest derivative present is \(\frac{d^2y}{dx^2}\), so
\(m = 2\)
It appears with power 2 (the cube is on the lower derivative \(\frac{dy}{dx}\)), so
\(n = 2\)
The general solution of a differential equation of order m contains m arbitrary constants.
So m = 2, n = 2, and the general solution has 2 arbitrary constants.
Checking the statements
(A) \(m = 2\) — Correct. The highest derivative is the second.
(B) \(n = 3\) — Incorrect. The degree is 2, the power of \(\frac{d^2y}{dx^2}\).
(C) The general solution contains two arbitrary constants — Correct. The order is 2.
(D) \(m = n\) — Correct. Both are 2.