The differential equation of the family of curves y = Ax + displaysty…

The correct answer is Option (1) → \(x^2\displaystyle\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = 0\)

There are two arbitrary constants, so the equation is of order 2: differentiate twice and eliminate A and B.

    \( \displaystyle \frac{dy}{dx} = A - \frac{B}{x^2} \)

    \( \displaystyle \frac{d^2y}{dx^2} = \frac{2B}{x^3} \)

Multiplying the first by x and subtracting from y removes A:

    \( \displaystyle x\frac{dy}{dx} = Ax - \frac{B}{x} \)

    \( \displaystyle y - x\frac{dy}{dx} = \frac{2B}{x} \)

Multiplying the second derivative by \(x^2\) gives the same quantity:

    \( \displaystyle x^2\frac{d^2y}{dx^2} = \frac{2B}{x} \)

So

    \( \displaystyle x^2\frac{d^2y}{dx^2} = y - x\frac{dy}{dx} \)

    \( \displaystyle x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = 0 \)