The general solution of the differential equation `(dy)/(dx)=2y tan x+tan^(2)x, AA x in (0, (pi)/(2))` is `yf(x)=(x)/(2)-(sin(2x))/(4)+C`, (where, C is an arbitrary constant). If `((pi)/(4))=(1)/(2)`, then the value of `f((pi)/(3))` is equal to

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. 2
D. 4

Answer

To solve the problem step by step, we will first analyze the given differential equation and its general solution. Then, we will find the value of \( f\left(\frac{\pi}{3}\right) \) based on the provided information.

### Step 1: Understand the General Solution
The general solution of the differential equation is given as:
\[ y_f(x) = \frac{x}{2} - \frac{\sin(2x)}{4} + C \]
where \( C \) is an arbitrary constant.

### Step 2: Use the Given Condition
We are given that:
\[ f\left(\frac{\pi}{4}\right) = \frac{1}{2} \]
We will substitute \( x = \frac{\pi}{4} \) into the general solution to find \( C \).

### Step 3: Substitute \( x = \frac{\pi}{4} \)
Substituting \( x = \frac{\pi}{4} \) into the general solution:
\[ f\left(\frac{\pi}{4}\right) = \frac{\frac{\pi}{4}}{2} - \frac{\sin\left(2 \cdot \frac{\pi}{4}\right)}{4} + C \]
Calculating \( \sin\left(2 \cdot \frac{\pi}{4}\right) \):
\[ \sin\left(\frac{\pi}{2}\right) = 1 \]
Thus, we have:
\[ f\left(\frac{\pi}{4}\right) = \frac{\pi}{8} - \frac{1}{4} + C \]

### Step 4: Set the Equation Equal to \( \frac{1}{2} \)
Now we set the expression equal to \( \frac{1}{2} \):
\[ \frac{\pi}{8} - \frac{1}{4} + C = \frac{1}{2} \]

### Step 5: Solve for \( C \)
Rearranging the equation to solve for \( C \):
\[ C = \frac{1}{2} - \frac{\pi}{8} + \frac{1}{4} \]
Converting \( \frac{1}{4} \) to eighths:
\[ \frac{1}{4} = \frac{2}{8} \]
Thus, we have:
\[ C = \frac{1}{2} - \frac{\pi}{8} + \frac{2}{8} = \frac{4}{8} - \frac{\pi}{8} = \frac{4 - \pi}{8} \]

### Step 6: Substitute \( C \) Back into the General Solution
Now, we substitute \( C \) back into the general solution:
\[ y_f(x) = \frac{x}{2} - \frac{\sin(2x)}{4} + \frac{4 - \pi}{8} \]

### Step 7: Find \( f\left(\frac{\pi}{3}\right) \)
Now we need to find \( f\left(\frac{\pi}{3}\right) \):
\[ f\left(\frac{\pi}{3}\right) = \frac{\frac{\pi}{3}}{2} - \frac{\sin\left(2 \cdot \frac{\pi}{3}\right)}{4} + \frac{4 - \pi}{8} \]
Calculating \( \sin\left(2 \cdot \frac{\pi}{3}\right) \):
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Substituting this value:
\[ f\left(\frac{\pi}{3}\right) = \frac{\pi}{6} - \frac{\frac{\sqrt{3}}{2}}{4} + \frac{4 - \pi}{8} \]
Simplifying:
\[ = \frac{\pi}{6} - \frac{\sqrt{3}}{8} + \frac{4 - \pi}{8} \]
Combining the terms:
\[ = \frac{\pi}{6} + \frac{4 - \pi - \sqrt{3}}{8} \]

### Step 8: Final Calculation
To find a common denominator (24):
\[ = \frac{4\pi}{24} + \frac{3(4 - \pi - \sqrt{3})}{24} = \frac{4\pi + 12 - 3\pi - 3\sqrt{3}}{24} = \frac{\pi + 12 - 3\sqrt{3}}{24} \]

### Conclusion
Thus, the value of \( f\left(\frac{\pi}{3}\right) \) is:
\[ f\left(\frac{\pi}{3}\right) = \frac{\pi + 12 - 3\sqrt{3}}{24} \]