If cos `x (dy)/(dx)-y sin x = 6x, (0 lt x lt (pi)/(2))` and `y((pi)/(3))=0`, then `y((pi)/(6))` is equal to :-

Options

A. (a) `- (pi^(2))/(4 sqrt(3))`
B. (b) `- (pi^(2))/(2)`
C. (c) `- (pi^(2))/(2sqrt(3))`
D. (d) `(pi^(2))/(2 sqrt(3))`

Explanation

To solve the differential equation given in the problem, we will follow these steps:

### Step 1: Rewrite the differential equation
The given differential equation is:
\[ \cos x \frac{dy}{dx} - y \sin x = 6x \]
We can divide the entire equation by \(\cos x\) to simplify it:
\[ \frac{dy}{dx} - y \tan x = 6x \sec x \]

### Step 2: Identify \(p\) and \(q\)
In the standard form \(\frac{dy}{dx} + p y = q\), we identify:
- \(p = -\tan x\)
- \(q = 6x \sec x\)

### Step 3: Find the integrating factor
The integrating factor \(I\) is given by:
\[ I = e^{\int p \, dx} = e^{\int -\tan x \, dx} \]
The integral of \(-\tan x\) is:
\[ -\ln |\sec x| = \ln |\cos x| \]
Thus, the integrating factor is:
\[ I = e^{\ln |\cos x|} = \cos x \]

### Step 4: Multiply the entire equation by the integrating factor
Multiplying the differential equation by \(\cos x\):
\[ \cos x \frac{dy}{dx} - y \sin x = 6x \]
This simplifies to:
\[ \frac{d}{dx}(y \cos x) = 6x \]

### Step 5: Integrate both sides
Integrating both sides with respect to \(x\):
\[ \int \frac{d}{dx}(y \cos x) \, dx = \int 6x \, dx \]
This gives:
\[ y \cos x = 3x^2 + C \]
where \(C\) is the constant of integration.

### Step 6: Solve for \(y\)
Rearranging gives:
\[ y = \frac{3x^2 + C}{\cos x} \]

### Step 7: Use the initial condition to find \(C\)
We are given the condition \(y\left(\frac{\pi}{3}\right) = 0\):
\[ 0 = \frac{3\left(\frac{\pi}{3}\right)^2 + C}{\cos\left(\frac{\pi}{3}\right)} \]
Since \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\):
\[ 0 = 3\left(\frac{\pi^2}{9}\right) + C \implies C = -\frac{\pi^2}{3} \]

### Step 8: Substitute \(C\) back into the equation for \(y\)
Substituting \(C\) back, we have:
\[ y = \frac{3x^2 - \frac{\pi^2}{3}}{\cos x} \]

### Step 9: Find \(y\left(\frac{\pi}{6}\right)\)
Now we need to find \(y\left(\frac{\pi}{6}\right)\):
\[ y\left(\frac{\pi}{6}\right) = \frac{3\left(\frac{\pi}{6}\right)^2 - \frac{\pi^2}{3}}{\cos\left(\frac{\pi}{6}\right)} \]
Calculating:
\[ y\left(\frac{\pi}{6}\right) = \frac{3\left(\frac{\pi^2}{36}\right) - \frac{\pi^2}{3}}{\frac{\sqrt{3}}{2}} = \frac{\frac{\pi^2}{12} - \frac{12\pi^2}{36}}{\frac{\sqrt{3}}{2}} = \frac{\frac{\pi^2}{12} - \frac{\pi^2}{3}}{\frac{\sqrt{3}}{2}} = \frac{\frac{\pi^2}{12} - \frac{4\pi^2}{12}}{\frac{\sqrt{3}}{2}} = \frac{-\frac{3\pi^2}{12}}{\frac{\sqrt{3}}{2}} = \frac{-\frac{\pi^2}{4}}{\frac{\sqrt{3}}{2}} = -\frac{\pi^2}{2\sqrt{3}} \]

### Final Answer
Thus, the value of \(y\left(\frac{\pi}{6}\right)\) is:
\[ y\left(\frac{\pi}{6}\right) = -\frac{\pi^2}{2\sqrt{3}} \]