`sin x cos y+(dy)/(dx)cos x sin y=0`

Options

A. `sin(x+y)=c`
B. `sec x sec y =c`
C. `sin x cos y=c`
D. `cos (x+y)=c`

Answer

To solve the differential equation \( \sin x \cos y + \frac{dy}{dx} \cos x \sin y = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start by isolating \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} \cos x \sin y = -\sin x \cos y \]
Now, divide both sides by \( \cos x \sin y \) (assuming \( \cos x \sin y \neq 0 \)):
\[ \frac{dy}{dx} = -\frac{\sin x \cos y}{\cos x \sin y} \]

### Step 2: Separating Variables
We can separate the variables \( y \) and \( x \):
\[ \frac{dy}{\sin y} = -\frac{\sin x}{\cos x} dx \]
This simplifies to:
\[ \frac{dy}{\sin y} = -\tan x \, dx \]

### Step 3: Integrating Both Sides
Now we integrate both sides:
\[ \int \frac{dy}{\sin y} = -\int \tan x \, dx \]
The integral of \( \frac{1}{\sin y} \) is \( \log |\csc y - \cot y| \) and the integral of \( \tan x \) is \( -\log |\cos x| \):
\[ \log |\csc y - \cot y| = -\log |\cos x| + C \]

### Step 4: Simplifying the Equation
We can rewrite the equation:
\[ \log |\csc y - \cot y| + \log |\cos x| = C \]
Using the property of logarithms, we combine the logs:
\[ \log |\cos x (\csc y - \cot y)| = C \]
Exponentiating both sides gives:
\[ |\cos x (\csc y - \cot y)| = e^C \]
Let \( k = e^C \), so we have:
\[ \cos x (\csc y - \cot y) = k \]

### Step 5: Final Form
We can express this in a more familiar form:
\[ \cos x \left( \frac{1 - \cos y}{\sin y} \right) = k \]
This can be rearranged to find a relationship between \( x \) and \( y \).

### Final Answer
Thus, the solution to the differential equation is:
\[ \sin x \sin y = C \]
where \( C \) is a constant.

---