For the differential equation (x + y) dy + (x - y) dx = 0 , which of…

The correct answer is Option (4) → (A) and (B) only

Writing the equation in derivative form:

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

So \(\frac{dy}{dx} = \frac{y - x}{x + y}\): only the first derivative appears, to the power 1.

Checking the statements

(A) Differential equation is homogeneous — Correct. The right side \(f(x, y) = \frac{y - x}{x + y}\) is a homogeneous function of degree zero:

    \(\displaystyle f(\lambda x, \lambda y) = \frac{\lambda y - \lambda x}{\lambda x + \lambda y} = f(x, y)\)

(B) Order of differential equation is 1 — Correct. The highest derivative present is \(\frac{dy}{dx}\).

(C) Integrating factor of differential equation is \(e^x\) — Incorrect. An integrating factor \(e^{\int P\,dx}\) belongs to a linear equation \(\frac{dy}{dx} + Py = Q\). Here y appears in the denominator, so the equation is not linear in y; it is solved as a homogeneous equation by putting y = vx.

(D) Degree of the equation is not defined — Incorrect. The equation is a polynomial in \(\frac{dy}{dx}\), which appears to the power 1, so its degree is 1.