Solution of the differential equation sin x. cos y dy + cos x. sin y dx = 0 is

Options

A. sin x + sin y = C
B. cos x + cos y = C
C. sin x . Sin y = C
D. `(sin x)/(sin y) = C`

Answer

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

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ \sin x \cos y \, dy = -\cos x \sin y \, dx \]

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

### Step 3: Integrating Both Sides
Now we integrate both sides:
\[ \int \cot y \, dy = -\int \cot x \, dx \]
The integral of \(\cot\) is:
\[ \ln |\sin y| = -\ln |\sin x| + C \]
where \(C\) is the constant of integration.

### Step 4: Simplifying the Equation
We can rewrite the equation as:
\[ \ln |\sin y| + \ln |\sin x| = C \]
Using the property of logarithms that states \(\ln a + \ln b = \ln(ab)\), we can combine the logs:
\[ \ln |\sin y \sin x| = C \]

### Step 5: Exponentiating Both Sides
To eliminate the logarithm, we exponentiate both sides:
\[ |\sin y \sin x| = e^C \]
Let \(k = e^C\), where \(k\) is a positive constant:
\[ \sin y \sin x = k \]

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

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