Consider the differential equation, $x\frac{dy}{dx}=y(\log_e y-\log_e x + 1)$, then which of the following are true?

(A) It is a linear differential equation
(B) It is a homogenous differential equation
(C) Its general solution is $\log_e (\frac{y}{x}) =Cx$, where C is constant of integration
(D) Its general solution is $\log_e(\frac{x}{y})= Cy$, where C is constant of integration
(E) If $y(1) = 1$, then its particular solution is $y = x$

Choose the correct answer from the options given below:

(B), (C) and (E) only

The correct answer is Option (4) → (B), (C) and (E) only

Given differential equation:

$x\frac{dy}{dx}=y(\log y-\log x+1)$

Rewrite:

$\frac{1}{y}\frac{dy}{dx}=\frac{\log y-\log x+1}{x}$

Substitute $u=\frac{y}{x}$, so $y=ux$ and

$\frac{dy}{dx}=u+x\frac{du}{dx}$

Substitute into the differential equation:

$x(u+x\frac{du}{dx})=ux(\log u+1)$

$xu+x^{2}\frac{du}{dx}=ux\log u+ux$

$x^{2}\frac{du}{dx}=ux\log u$

$\frac{du}{dx}=\frac{u\log u}{x}$

$\frac{du}{u\log u}=\frac{dx}{x}$

Integrate:

$\int \frac{du}{u\log u}=\int \frac{dx}{x}$

$\log (\log u)=\log x + C$

$\log \left(\log \frac{y}{x}\right)=\log x + C$

This matches option (C).

Check linearity: equation is nonlinear → (A) false.

Check homogeneity: it reduces to a function of $\frac{y}{x}$ → (B) true.

Check particular solution for $y(1)=1$:

$u=\frac{y}{x}=1$ gives $\log 1 =0$ satisfying the equation, so $y=x$ is valid → (E) true.

Final answer: (B), (C), and (E) are correct