Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = \frac{\tan x + y}{\sin x(\sec x - \sin x \tan x)}, x \in \left(0, \frac{\pi}{2}\right)$$ satisfying the condition $$y\left(\frac{\pi}{4}\right) = 2$$. Then, $$y\left(\frac{\pi}{3}\right)$$ is

The differential equation is $$\frac{dy}{dx} = \frac{\tan x + y}{\sin x(\sec x - \sin x\tan x)}$$, with $$y(\pi/4) = 2$$.

Simplify the denominator: $$\sin x(\sec x - \sin x\tan x) = \sin x \cdot \sec x - \sin^2 x \cdot \tan x$$

$$= \frac{\sin x}{\cos x} - \frac{\sin^3 x}{\cos x} = \frac{\sin x(1 - \sin^2 x)}{\cos x} = \frac{\sin x \cos^2 x}{\cos x} = \sin x\cos x$$

So the equation becomes: $$\frac{dy}{dx} = \frac{\tan x + y}{\sin x\cos x}$$

$$\sin x\cos x \cdot \frac{dy}{dx} = \tan x + y$$

$$\sin x\cos x \cdot \frac{dy}{dx} - y = \tan x$$

$$\frac{dy}{dx} - \frac{y}{\sin x\cos x} = \frac{\tan x}{\sin x\cos x} = \frac{1}{\cos^2 x} = \sec^2 x$$

This is a linear first-order ODE: $$\frac{dy}{dx} + P(x)y = Q(x)$$ where $$P = -\frac{1}{\sin x\cos x} = -\frac{2}{\sin 2x}$$ and $$Q = \sec^2 x$$.

Integrating factor: $$\mu = e^{\int P\,dx} = e^{-\int \frac{2}{\sin 2x}dx} = e^{-\int \csc 2x \cdot 2\,dx}$$.

$$\int \csc 2x \cdot 2\,dx$$: Let $$u = 2x$$, then $$\int \csc u\,du = \ln|\csc u - \cot u| = \ln\left|\tan\frac{u}{2}\right| = \ln|\tan x|$$.

So $$\mu = e^{-\ln|\tan x|} = \frac{1}{\tan x} = \cot x$$.

Multiplying: $$\frac{d}{dx}(y\cot x) = \sec^2 x \cdot \cot x = \frac{1}{\sin x\cos x} = \frac{2}{\sin 2x}$$.

$$y\cot x = \int \frac{2}{\sin 2x}dx = \int \csc 2x \cdot 2 \cdot \frac{dx}{1}$$... Using the result above: $$\int \frac{2}{\sin 2x}dx = \ln|\tan x| + C$$.

So $$y\cot x = \ln|\tan x| + C$$, i.e., $$y = \tan x(\ln|\tan x| + C)$$.

Using $$y(\pi/4) = 2$$: $$2 = \tan(\pi/4)(\ln|\tan(\pi/4)| + C) = 1 \cdot (0 + C)$$, so $$C = 2$$.

Therefore $$y = \tan x(\ln|\tan x| + 2)$$.

At $$x = \pi/3$$: $$y(\pi/3) = \tan(\pi/3)(\ln|\tan(\pi/3)| + 2) = \sqrt{3}(\ln\sqrt{3} + 2) = \sqrt{3}\left(\frac{1}{2}\ln 3 + 2\right)$$

$$= \sqrt{3}(2 + \log_e\sqrt{3})$$

The correct answer is Option A: $$\sqrt{3}(2 + \log_e\sqrt{3})$$.