Let $$y = y(x)$$ be the solution of the differential equation $$\sin x \frac{dy}{dx} + y \cos x = 4x$$, $$x \in (0, \pi)$$. If $$y\left(\frac{\pi}{2}\right) = 0$$, then $$y\left(\frac{\pi}{6}\right)$$ is equal to:

We start with the given first-order linear differential equation

$$\sin x\,\frac{dy}{dx}+y\cos x = 4x,\qquad x\in(0,\pi).$$

In order to recognise it in the standard linear form, we divide every term by $$\sin x\;( \neq 0\text{ for }x\in(0,\pi))$$. This gives

$$\frac{dy}{dx}+\frac{\cos x}{\sin x}\,y = \frac{4x}{\sin x}.$$

The fraction $$\dfrac{\cos x}{\sin x}$$ is the trigonometric function $$\cot x$$ and $$\dfrac{1}{\sin x}$$ is $$\csc x$$, so we can rewrite the equation compactly as

$$\frac{dy}{dx}+y\cot x = 4x\csc x.$$

For a linear differential equation of the form

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

the integrating factor (I.F.) is defined by the formula

$$\text{I.F.}=e^{\displaystyle\int P(x)\,dx}.$$

Here $$P(x)=\cot x$$, so we compute

$$\int P(x)\,dx=\int\cot x\,dx=\int\frac{\cos x}{\sin x}\,dx=\ln|\sin x|.$$

Exponentiating, we obtain the integrating factor

$$\text{I.F.}=e^{\ln|\sin x|}=|\sin x|.$$

Because $$x\in(0,\pi)$$, we have $$\sin x>0$$ and therefore the absolute value can be dropped, giving simply

$$\text{I.F.}= \sin x.$$

We multiply the whole differential equation by this integrating factor:

$$\sin x\left(\frac{dy}{dx}\right)+\sin x\cdot y\cot x = \sin x\cdot 4x\csc x.$$

The right-hand side simplifies because $$\sin x\cdot\csc x = 1$$:

$$\sin x\,\frac{dy}{dx}+y(\sin x\cot x) = 4x.$$

Recalling that $$\cot x = \dfrac{\cos x}{\sin x}$$, we have $$\sin x\cot x = \cos x$$, so the left-hand side is

$$\sin x\,\frac{dy}{dx}+y\cos x.$$

Thus the equation has become

$$\sin x\,\frac{dy}{dx}+y\cos x = 4x,$$

which is in fact the same expression we started with, confirming that we have used the correct integrating factor.

By construction of the integrating factor, the left-hand side now represents the derivative of the product $$y\sin x$$, because

$$\frac{d}{dx}(y\sin x)=\sin x\,\frac{dy}{dx}+y\cos x.$$

Therefore we can rewrite the differential equation compactly as

$$\frac{d}{dx}(y\sin x)=4x.$$

We integrate both sides with respect to $$x$$:

$$\int\frac{d}{dx}(y\sin x)\,dx = \int 4x\,dx.$$

The integral on the left simply returns the product $$y\sin x$$, while the integral on the right is

$$\int 4x\,dx = 4\cdot\frac{x^{2}}{2}=2x^{2}.$$

Adding the constant of integration $$C$$, we have

$$y\sin x = 2x^{2}+C.$$

Solving for $$y$$ gives the general solution

$$y(x)=\frac{2x^{2}+C}{\sin x}.$$

We now use the initial condition $$y\!\left(\dfrac{\pi}{2}\right)=0$$ to find the constant $$C$$. Substituting $$x=\dfrac{\pi}{2}$$, and recalling that $$\sin\dfrac{\pi}{2}=1$$, we get

$$0 = y\!\left(\dfrac{\pi}{2}\right)=\frac{2\left(\dfrac{\pi}{2}\right)^{2}+C}{1}.$$

Simplifying inside the numerator,

$$2\left(\dfrac{\pi}{2}\right)^{2}=2\cdot\frac{\pi^{2}}{4}=\frac{\pi^{2}}{2}.$$

Thus

$$0 = \frac{\pi^{2}}{2}+C \quad\Longrightarrow\quad C = -\frac{\pi^{2}}{2}.$$

Substituting this value of $$C$$ back into the expression for $$y(x)$$, we obtain

$$y(x)=\frac{2x^{2}-\dfrac{\pi^{2}}{2}}{\sin x}.$$

For later convenience we multiply numerator and denominator by $$2$$, giving an equivalent but slightly tidier form:

$$y(x)=\frac{4x^{2}-\pi^{2}}{2\sin x}.$$

Now we evaluate $$y\!\left(\dfrac{\pi}{6}\right)$$. First, $$\sin\dfrac{\pi}{6}=\dfrac12$$. Substituting $$x=\dfrac{\pi}{6}$$ into the numerator, we compute

$$4x^{2}-\pi^{2}=4\left(\frac{\pi}{6}\right)^{2}-\pi^{2}=4\cdot\frac{\pi^{2}}{36}-\pi^{2}=\frac{\pi^{2}}{9}-\pi^{2}.$$

Hence

$$y\!\left(\frac{\pi}{6}\right)=\frac{\dfrac{\pi^{2}}{9}-\pi^{2}}{2\cdot\dfrac12}.$$

The denominator $$2\cdot\dfrac12$$ equals $$1$$, so it disappears, leaving simply

$$y\!\left(\frac{\pi}{6}\right)=\frac{\pi^{2}}{9}-\pi^{2}.$$

Combining the terms inside the numerator by taking a common denominator of $$9$$, we obtain

$$\frac{\pi^{2}}{9}-\pi^{2}=\frac{\pi^{2}}{9}-\frac{9\pi^{2}}{9}=\frac{-8\pi^{2}}{9}.$$

Therefore

$$y\!\left(\frac{\pi}{6}\right)=-\frac{8}{9}\pi^{2}.$$

Among the given options, this value corresponds to Option D.

Hence, the correct answer is Option D.