The solution of the differential equation
`(dy)/(dx)=(siny+x)/(sin2y-x cos y)` is

Options

A. `sin^(2)y=x sin y+(x^(2))/(2)+c`
B. `sin^(2)y=x siny-(x^(2))/(2)+c`
C. `sin^(2)y=x+siny+(x^(2))/(2)+c`
D. `sin^(2)y=x-siny+(x^(2))/(2)+c`

Answer

To solve the differential equation

\[ \frac{dy}{dx} = \frac{\sin y + x}{\sin 2y - x \cos y} \]

we will follow these steps:

### Step 1: Separate the variables
We can rearrange the equation to separate the variables \(y\) and \(x\):

\[ (\sin 2y - x \cos y) dy = (\sin y + x) dx \]

### Step 2: Rewrite the equation
Now, we can rewrite the equation as:

\[ \sin 2y \, dy - x \cos y \, dy = \sin y \, dx + x \, dx \]

### Step 3: Integrate both sides
Next, we will integrate both sides. The left side will be integrated with respect to \(y\) and the right side with respect to \(x\):

\[ \int \sin 2y \, dy - \int x \cos y \, dy = \int \sin y \, dx + \int x \, dx \]

### Step 4: Solve the integrals
1. **Left Side:**
- The integral of \(\sin 2y\) is \(-\frac{1}{2} \cos 2y\).
- The integral of \(-x \cos y \, dy\) can be treated as \(x \int \cos y \, dy = x \sin y\) (keeping \(x\) constant during integration).

2. **Right Side:**
- The integral of \(\sin y \, dx\) is \(x \sin y\) (keeping \(y\) constant).
- The integral of \(x \, dx\) is \(\frac{x^2}{2}\).

Putting it all together, we have:

\[ -\frac{1}{2} \cos 2y + x \sin y = x \sin y + \frac{x^2}{2} + C \]

### Step 5: Simplify the equation
We can simplify the equation by canceling \(x \sin y\) from both sides:

\[ -\frac{1}{2} \cos 2y = \frac{x^2}{2} + C \]

### Step 6: Rearranging the equation
To express \(C\) in terms of the other variables, we can rearrange it:

\[ \cos 2y = -x^2 - 2C \]

### Step 7: Final form
We can express the solution in a more standard form. Let \(C' = -2C\):

\[ \cos 2y = -x^2 + C' \]

This gives us the general solution to the differential equation.