Consider the differential equation xdy = (x + y) dx. Which of the fol…

The correct answer is Option (3) → (A) and (D) only

Given: $x\,dy = (x + y)\,dx$

Rewriting: $x \frac{dy}{dx} = x + y$

$\Rightarrow x \frac{dy}{dx} - y = x$

This is a linear differential equation of the form:

$\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x) = -\frac{1}{x}$ and $Q(x) = 1$

Integrating Factor (I.F.) $= e^{\int P(x) dx} = e^{\int -\frac{1}{x} dx} = e^{-\ln|x|} = \frac{1}{x}$

Multiply entire equation by $\frac{1}{x}$:

$\frac{1}{x} \cdot \frac{dy}{dx} - \frac{y}{x^2} = 1/x$

$\Rightarrow \frac{d}{dx} \left(\frac{y}{x}\right) = \frac{1}{x}$

Integrate both sides:

$\frac{y}{x} = \ln|x| + C \Rightarrow y = x\ln|x| + Cx$

Now verify the statements:

(A) ✔️ True — $\frac{dy}{dx} = 1 + \frac{y}{x}$ is homogeneous

(B) ❌ False — Order = 1

(C) ❌ False — Only 1 arbitrary constant

(D) ✔️ True — I.F. = $\frac{1}{x}$

(E) ❌ False — Degree is defined and equals 1