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 $\log_e\left(\frac{y}{x}\right) = Cx$, where C is constant of integration

(D) Its general solution is $\log_e\left(\frac{x}{y}\right) = Cy$, 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:

The differential equation is $x\frac{dy}{dx} = y(\log_e y - \log_e x + 1)$.

Rewrite using $\log_e y - \log_e x = \log_e\left(\frac{y}{x}\right)$:

$x\frac{dy}{dx} = y\left(\log_e\left(\frac{y}{x}\right) + 1\right)$

$\frac{dy}{dx} = \frac{y}{x}\left(\log_e\left(\frac{y}{x}\right) + 1\right)$

---

A linear differential equation has the form $\frac{dy}{dx} + P(x)y = Q(x)$ where $y$ should not appear inside any function.

The equation contains $\log_e y$, so $y$ appears inside a logarithm.

Statement (A) is FALSE.

---

A homogeneous differential equation can be written as $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$.

The equation $\frac{dy}{dx} = \frac{y}{x}\left(\log_e\left(\frac{y}{x}\right) + 1\right)$ is a function of $\frac{y}{x}$ only.

Statement (B) is TRUE.

---

Let $v = \frac{y}{x}$, so $y = vx$ and $\frac{dy}{dx} = v + x\frac{dv}{dx}$.

$v + x\frac{dv}{dx} = v(\log_e v + 1)$

$x\frac{dv}{dx} = v\log_e v + v - v$

$x\frac{dv}{dx} = v\log_e v$

$\frac{dv}{v\log_e v} = \frac{dx}{x}$

---

Let $u = \log_e v$, then $du = \frac{dv}{v}$.

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

Integrating both sides:

$\log_e|u| = \log_e|x| + \log_e|C|$

$u = Cx$

---

Since $u = \log_e v$:

$\log_e v = Cx$

Since $v = \frac{y}{x}$:

$\log_e\left(\frac{y}{x}\right) = Cx$

Statement (C) is TRUE.

Statement (D) is FALSE.

---

For the initial condition $y(1) = 1$:

$\log_e\left(\frac{1}{1}\right) = C(1)$

$\log_e(1) = C$

$0 = C$

So $C = 0$:

$\log_e\left(\frac{y}{x}\right) = 0$

$\frac{y}{x} = e^0$

$\frac{y}{x} = 1$

$y = x$

Statement (E) is TRUE.

---

True statements: (B), (C), and (E)