What is the equation of the curve passing through the point ( 0 , `(pi)/(3)`) satisfying the differential equation sin x cos y dx + cos x sin y dy = 0 ?

Options

A. `cos x cos y = (sqrt3)/(2)`
B. `sin x sin y = (sqrt3)/(2)`
C. `sin x sin y = (1)/(2)`
D. `cos x cos y = (1)/(2)`

Answer

To solve the given differential equation \( \sin x \cos y \, dx + \cos x \sin y \, dy = 0 \) and find the equation of the curve passing through the point \( (0, \frac{\pi}{3}) \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ \sin x \cos y \, dx + \cos x \sin y \, dy = 0 \]
We can rearrange this to separate the variables:
\[ \sin x \cos y \, dx = -\cos x \sin y \, dy \]
Dividing both sides by \( \sin y \cos y \) (assuming \( \sin y \cos y \neq 0 \)):
\[ \frac{\sin x}{\sin y} \, dx = -\frac{\cos x}{\cos y} \, dy \]

### Step 2: Separating Variables
Now we can separate the variables:
\[ \frac{\sin x}{\cos x} \, dx = -\frac{\sin y}{\cos y} \, dy \]
This can be rewritten as:
\[ \tan x \, dx = -\tan y \, dy \]

### Step 3: Integrating Both Sides
Next, we integrate both sides:
\[ \int \tan x \, dx = -\int \tan y \, dy \]
The integral of \( \tan x \) is \( -\ln |\cos x| \), and similarly for \( y \):
\[ -\ln |\cos x| = -(-\ln |\cos y|) + C \]
This simplifies to:
\[ \ln |\cos x| = \ln |\cos y| + C \]

### Step 4: Exponentiating Both Sides
Exponentiating both sides gives:
\[ |\cos x| = k |\cos y| \]
where \( k = e^C \) is a constant.

### Step 5: Finding the Constant Using Initial Condition
We need to find the value of \( k \) using the initial condition that the curve passes through the point \( (0, \frac{\pi}{3}) \):
\[ |\cos(0)| = k |\cos(\frac{\pi}{3})| \]
Calculating the cosine values:
\[ 1 = k \cdot \frac{1}{2} \]
Thus, we find:
\[ k = 2 \]

### Step 6: Writing the Final Equation
Substituting \( k \) back into the equation:
\[ |\cos x| = 2 |\cos y| \]
This can be simplified to:
\[ \cos x = 2 \cos y \]
This is the equation of the curve that satisfies the given differential equation and passes through the specified point.

### Final Answer
The equation of the curve is:
\[ \cos x = 2 \cos y \]