Solution of the differential equation
`sin2x (dy)/(dx) -y=tan x` is

Options

A. `y=tan x +c sqrt(tan x)`
B. `x-y sin x=c`
C. `xy tan x =c`
D. none of these

Answer

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

### Step 1: Rearranging the Equation
First, we rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ \sin(2x) \frac{dy}{dx} = y + \tan(x) \]
Now, divide both sides by \( \sin(2x) \):
\[ \frac{dy}{dx} = \frac{y}{\sin(2x)} + \frac{\tan(x)}{\sin(2x)} \]

### Step 2: Identify the Standard Form
The equation can be rewritten in the standard form of a linear differential equation:
\[ \frac{dy}{dx} + P(x) y = Q(x) \]
where \( P(x) = -\frac{1}{\sin(2x)} \) and \( Q(x) = \frac{\tan(x)}{\sin(2x)} \).

### Step 3: Finding the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int -\frac{1}{\sin(2x)} \, dx} \]
To compute this integral, we can use the identity \( \sin(2x) = 2 \sin(x) \cos(x) \):
\[ \int -\frac{1}{\sin(2x)} \, dx = -\frac{1}{2} \int \left( \frac{1}{\sin(x)} + \frac{1}{\cos(x)} \right) \, dx \]
This integral can be computed, leading to:
\[ \mu(x) = e^{-\frac{1}{2} \log(\sin(2x))} = \frac{1}{\sqrt{\sin(2x)}} \]

### Step 4: Multiply the Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor:
\[ \frac{1}{\sqrt{\sin(2x)}} \frac{dy}{dx} - \frac{y}{\sin(2x)^{3/2}} = \frac{\tan(x)}{\sin(2x)^{3/2}} \]

### Step 5: Solve the Left Side
The left-hand side can be expressed as:
\[ \frac{d}{dx} \left( y \cdot \frac{1}{\sqrt{\sin(2x)}} \right) \]
Thus, we have:
\[ \frac{d}{dx} \left( y \cdot \frac{1}{\sqrt{\sin(2x)}} \right) = \frac{\tan(x)}{\sin(2x)^{3/2}} \]

### Step 6: Integrate Both Sides
Integrating both sides gives:
\[ y \cdot \frac{1}{\sqrt{\sin(2x)}} = \int \frac{\tan(x)}{\sin(2x)^{3/2}} \, dx + C \]

### Step 7: Solve for y
Finally, we solve for \( y \):
\[ y = \sqrt{\sin(2x)} \left( \int \frac{\tan(x)}{\sin(2x)^{3/2}} \, dx + C \right) \]

### Final Solution
The final solution to the differential equation is:
\[ y = \sqrt{\sin(2x)} \left( \int \frac{\tan(x)}{\sin(2x)^{3/2}} \, dx + C \right) \]