If `y=y(x)` is the solution curve of the differential equation `(dy)/(dx)+y tan x=x sec x,0lexle(pi)/(3),y(0)=1`, then `y((pi)/(6))` is equal to

Options

A. `(pi)/(12)-sqrt(3)/2log_(e)((2)/(esqrt(3)))`
B. `(pi)/(12)+sqrt(3)/2log_(e)((2)/(esqrt(3)))`
C. `(pi)/(12)+sqrt(3)/2log_(e)((2sqrt3)/(e))`
D. `(pi)/(12)-sqrt(3)/2log_(e)((2sqrt3)/(e))`

Answer

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

### Step 1: Identify the Differential Equation
The differential equation is given as:
\[ \frac{dy}{dx} + y \tan x = x \sec x \]
This is a first-order linear ordinary differential equation of the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \( P(x) = \tan x \) and \( Q(x) = x \sec x \).

### Step 2: Find the Integrating Factor
The integrating factor \( \mu(x) \) is calculated using:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \tan x \, dx} \]
The integral of \( \tan x \) is:
\[ \int \tan x \, dx = -\ln(\cos x) = \ln(\sec x) \]
Thus, the integrating factor becomes:
\[ \mu(x) = e^{\ln(\sec x)} = \sec x \]

### Step 3: Multiply the Differential Equation by the Integrating Factor
Now, we multiply the entire differential equation by \( \sec x \):
\[ \sec x \frac{dy}{dx} + y \sec x \tan x = x \]

### Step 4: Rewrite the Left Side as a Derivative
The left side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(y \sec x) = x \]

### Step 5: Integrate Both Sides
Integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(y \sec x) \, dx = \int x \, dx \]
This gives:
\[ y \sec x = \frac{x^2}{2} + C \]

### Step 6: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{x^2}{2 \sec x} + C \cos x \]

### Step 7: Use the Initial Condition
We are given the initial condition \( y(0) = 1 \):
\[ 1 = \frac{0^2}{2 \sec(0)} + C \cos(0) \]
Since \( \sec(0) = 1 \) and \( \cos(0) = 1 \), we have:
\[ 1 = 0 + C \cdot 1 \implies C = 1 \]
Thus, the solution becomes:
\[ y = \frac{x^2}{2 \sec x} + \cos x \]

### Step 8: Find \( y\left(\frac{\pi}{6}\right) \)
Now we need to evaluate \( y\left(\frac{\pi}{6}\right) \):
\[ y\left(\frac{\pi}{6}\right) = \frac{\left(\frac{\pi}{6}\right)^2}{2 \sec\left(\frac{\pi}{6}\right)} + \cos\left(\frac{\pi}{6}\right) \]
Calculating \( \sec\left(\frac{\pi}{6}\right) = \frac{1}{\cos\left(\frac{\pi}{6}\right)} = \frac{2}{\sqrt{3}} \) and \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \):
\[ y\left(\frac{\pi}{6}\right) = \frac{\left(\frac{\pi}{6}\right)^2}{2 \cdot \frac{2}{\sqrt{3}}} + \frac{\sqrt{3}}{2} \]
\[ = \frac{\left(\frac{\pi^2}{36}\right) \cdot \sqrt{3}}{4} + \frac{\sqrt{3}}{2} \]
\[ = \frac{\pi^2 \sqrt{3}}{144} + \frac{72\sqrt{3}}{144} = \frac{\pi^2 + 72}{144} \sqrt{3} \]

### Final Answer
Thus, the value of \( y\left(\frac{\pi}{6}\right) \) is:
\[ y\left(\frac{\pi}{6}\right) = \frac{\pi^2 + 72}{144} \sqrt{3} \]