Consider the following statements about differential equations.
(A) \(\displaystyle\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right)^2 = 0\) has order 2 and degree 1
(B) \(\left(\displaystyle\frac{dy}{dx}\right)^3 + y^4 = x\) has degree 4
(C) \(\displaystyle\frac{d^3y}{dx^3} + 4\frac{dy}{dx} = e^{x}\) has order 1
(D) \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^2 + \cos x = \left(\frac{dy}{dx}\right)^3\) has degree 2
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
The order is the order of the highest derivative; the degree is the power of that highest derivative. Powers of y or functions of x alone (such as \(y^4\) or \(\cos x\)) do not affect either.
(A): highest derivative \(\frac{d^2y}{dx^2}\), power 1
(B): highest derivative \(\frac{dy}{dx}\), power 3
(C): highest derivative \(\frac{d^3y}{dx^3}\), power 1
(D): highest derivative \(\frac{d^2y}{dx^2}\), power 2
So the (order, degree) pairs are (2, 1), (1, 3), (3, 1) and (2, 2).
Checking the statements
(A) order 2 and degree 1 — Correct. \(\frac{d^2y}{dx^2}\) appears with power 1.
(B) degree 4 — Incorrect. The degree is 3, the power of \(\frac{dy}{dx}\); the 4 is a power of y.
(C) order 1 — Incorrect. The order is 3, from \(\frac{d^3y}{dx^3}\).
(D) degree 2 — Correct. \(\frac{d^2y}{dx^2}\) appears squared; \(\cos x\) has no derivative in it.