Which of the following are correct?(A) The order of the differential…

The correct answer is Option (4) → (C) and (D) only

The order of a differential equation 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.

In (A), the highest derivative is \( \frac{dy}{dx} \), so the order is 1. In (B), the highest derivative \( \frac{d^2y}{dx^2} \) appears with power 1, so the degree is 1.

For (C), differentiate y = cx to remove c:

    \( \displaystyle \frac{dy}{dx}=c=\frac{y}{x} \)

    \( \displaystyle x\,dy=y\,dx \)

    \( \displaystyle y\,dx-x\,dy=0 \)

So the order in (A) is 1, the degree in (B) is 1, and y = cx gives y dx − x dy = 0.

Checking the statements

(A) The order of the differential equation \( \left(\frac{dy}{dx}\right)^2+3y^2-5y=0 \) is 2. — Incorrect. The order is 1; the 2 is the power of \( \frac{dy}{dx} \).

(B) The degree of the differential equation \( x\frac{d^2y}{dx^2}=\left(3+2\left(\frac{dy}{dx}\right)^2\right)^3 \) is 3. — Incorrect. The degree is 1; the power 3 is on an expression in the lower derivative \( \frac{dy}{dx} \).

(C) The differential equation representing the curve \( y=cx \), \( c \) being an arbitrary constant is \( ydx-xdy=0 \). — Correct. Removing c gives \( \frac{dy}{dx}=\frac{y}{x} \), i.e. y dx − x dy = 0.

(D) The general solution of an ordinary differential equation of n^th order contains n independent arbitrary constants. — Correct. This is the definition of the general solution: an equation of order n needs n integrations, each bringing one arbitrary constant.