sec xdy + cosec y dx =0

Answer

To solve the differential equation \( \sec x \, dy + \csc y \, dx = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sec x \, dy + \csc y \, dx = 0 \]
This can be rearranged to isolate \( dy \) and \( dx \):
\[ \sec x \, dy = -\csc y \, dx \]

### Step 2: Separate the variables
Next, we can separate the variables \( y \) and \( x \):
\[ \frac{dy}{\csc y} = -\frac{dx}{\sec x} \]
This simplifies to:
\[ \sin y \, dy = -\cos x \, dx \]

### Step 3: Integrate both sides
Now we will integrate both sides:
\[ \int \sin y \, dy = -\int \cos x \, dx \]
The integrals are:
\[ -\cos y = -\sin x + C \]
where \( C \) is the constant of integration.

### Step 4: Rearranging the equation
We can rearrange the equation to express it in a more standard form:
\[ \cos y + \sin x = C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ \cos y + \sin x = C \]