Solve `sin x (dy)/(dx)-y=sin x. tan""x/2`

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sin x \frac{dy}{dx} - y = \sin x \tan \frac{x}{2} \]

### Step 2: Divide by \(\sin x\)
Next, we divide the entire equation by \(\sin x\):
\[ \frac{dy}{dx} - \frac{y}{\sin x} = \tan \frac{x}{2} \]
This can also be written as:
\[ \frac{dy}{dx} - \csc x \cdot y = \tan \frac{x}{2} \]

### Step 3: Identify \(p\) and \(q\)
Now, we can identify \(p\) and \(q\) from the standard form of the linear differential equation:
\[ \frac{dy}{dx} + p y = q \]
Here, we have:
\[ p = -\csc x \quad \text{and} \quad q = \tan \frac{x}{2} \]

### Step 4: Find the integrating factor \(IF\)
The integrating factor \(IF\) is given by:
\[ IF = e^{\int p \, dx} = e^{\int -\csc x \, dx} \]
The integral of \(-\csc x\) is:
\[ \int -\csc x \, dx = -\ln |\tan \frac{x}{2}| \]
Thus,
\[ IF = e^{-\ln |\tan \frac{x}{2}|} = \frac{1}{\tan \frac{x}{2}} \]

### Step 5: Multiply the equation by the integrating factor
Now we multiply the entire differential equation by the integrating factor:
\[ \frac{1}{\tan \frac{x}{2}} \frac{dy}{dx} - \frac{y}{\tan \frac{x}{2} \sin x} = \frac{\tan \frac{x}{2}}{\tan \frac{x}{2}} \]
This simplifies to:
\[ \frac{1}{\tan \frac{x}{2}} \frac{dy}{dx} - \frac{y}{\tan \frac{x}{2}} = 1 \]

### Step 6: Integrate both sides
Now we can integrate both sides:
\[ \int \left( \frac{1}{\tan \frac{x}{2}} \frac{dy}{dx} - \frac{y}{\tan \frac{x}{2}} \right) dx = \int 1 \, dx \]
This gives us:
\[ y \cdot \cot \frac{x}{2} = x + C \]

### Step 7: Solve for \(y\)
Finally, we can express \(y\) in terms of \(x\):
\[ y = (x + C) \tan \frac{x}{2} \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = (x + C) \tan \frac{x}{2} \]