The general solution of `(dy)/(dx) =2ytan x+ tan ^(2)x` is:

Options

A. A) `y cos ^(2) x=x/2+(sin 2x)/(4)+C`
B. B) `y sec ^(2) x=x/2-(sin 2x)/(4)+C`
C. C) `y cos ^(2) x=x/2-(cos 2x)/(4)+C`
D. D) `y cos ^(2) x=x/2-(sin 2x)/(4)+C`

Answer

To solve the differential equation \(\frac{dy}{dx} = 2y \tan x + \tan^2 x\), we will follow these steps:

### Step 1: Rewrite the differential equation
We start by rewriting the given differential equation in standard form:
\[ \frac{dy}{dx} - 2y \tan x = \tan^2 x \]
This is in the form \(\frac{dy}{dx} + P(x)y = Q(x)\), where \(P(x) = -2 \tan x\) and \(Q(x) = \tan^2 x\).

**Hint:** Identify the coefficients of \(y\) and the non-homogeneous part of the equation.

### Step 2: Find the integrating factor
The integrating factor \(IF\) is given by:
\[ IF = e^{\int P(x) \, dx} = e^{\int -2 \tan x \, dx} \]
To compute the integral:
\[ \int -2 \tan x \, dx = -2 \ln |\cos x| = \ln |\cos^2 x|^{-1} \]
Thus, the integrating factor becomes:
\[ IF = e^{\ln |\cos^2 x|^{-1}} = \frac{1}{\cos^2 x} \]

**Hint:** Remember that the integrating factor is derived from the exponential of the integral of \(P(x)\).

### Step 3: Multiply the entire equation by the integrating factor
Now, we multiply the entire differential equation by the integrating factor:
\[ \frac{1}{\cos^2 x} \frac{dy}{dx} - \frac{2y \tan x}{\cos^2 x} = \frac{\tan^2 x}{\cos^2 x} \]
This simplifies to:
\[ \frac{1}{\cos^2 x} \frac{dy}{dx} - 2y \sec^2 x = \sec^2 x \]

**Hint:** Multiplying by the integrating factor transforms the left side into the derivative of a product.

### Step 4: Recognize the left side as a derivative
The left side can be recognized as:
\[ \frac{d}{dx}(y \sec^2 x) = \sec^2 x \]

**Hint:** Use the product rule to identify the left-hand side as a derivative.

### Step 5: Integrate both sides
Now we integrate both sides:
\[ \int \frac{d}{dx}(y \sec^2 x) \, dx = \int \sec^2 x \, dx \]
This gives:
\[ y \sec^2 x = \tan x + C \]

**Hint:** Remember that the integral of \(\sec^2 x\) is \(\tan x\).

### Step 6: Solve for \(y\)
Now we solve for \(y\):
\[ y = \tan x \cos^2 x + C \cos^2 x \]

**Hint:** Isolate \(y\) to express it in terms of \(x\) and \(C\).

### Step 7: Final form of the solution
We can express the solution as:
\[ y \cos^2 x = \frac{x}{2} - \frac{\sin 2x}{4} + C \]
Thus, the general solution is:
\[ y \cos^2 x = \frac{x}{2} - \frac{\sin 2x}{4} + C \]

**Hint:** Verify that the form matches one of the options given in the question.

### Conclusion
The correct answer is:
\[ \text{Option D: } y \cos^2 x = \frac{x}{2} - \frac{\sin 2x}{4} + C \]