Consider the following statements: (A) The general solution of a diff…

The correct answer is Option (2) → (A) and (B) only

The general solution contains as many arbitrary constants as the order of the equation. A particular solution is obtained from it by giving particular values to these constants, so it is free from arbitrary constants.

The degree is defined only when the equation is a polynomial in its derivatives. For example, \(\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0\) has no degree.

The family \(y = mx\) has one arbitrary constant. Differentiating once:

    \( \displaystyle \frac{dy}{dx} = m \)

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

So an order-3 equation has 3 constants in its general solution, a particular solution has none, the degree is not always defined, and the family y = mx gives an equation of order 1.

Checking the statements

(A) The general solution of a differential equation of order 3 contains 3 arbitrary constants. — Correct. The number of constants equals the order, 3.

(B) A particular solution of a differential equation contains no arbitrary constant. — Correct. All its constants have been given particular values.

(C) The degree of every differential equation is defined. — Incorrect. \(\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0\) is not a polynomial in the derivatives.

(D) The differential equation of the family of lines \(y = mx\), where m is an arbitrary constant, is of order 2. — Incorrect. One constant needs one differentiation: \(y = x\frac{dy}{dx}\), of order 1.