Solve: `(dy)/(dx)=secy`

Explanation

To solve the differential equation \(\frac{dy}{dx} = \sec y\), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ \frac{dy}{dx} = \sec y \]
This can be rewritten as:
\[ \frac{dx}{dy} = \frac{1}{\sec y} \]

### Step 2: Simplify the right-hand side
We know that \(\sec y = \frac{1}{\cos y}\), so:
\[ \frac{dx}{dy} = \cos y \]

### Step 3: Integrate both sides
Now, we will integrate both sides:
\[ \int dx = \int \cos y \, dy \]

### Step 4: Perform the integration
The left-hand side integrates to:
\[ x = \int dx = x + C_1 \]
The right-hand side integrates to:
\[ \int \cos y \, dy = \sin y + C_2 \]
Thus, we have:
\[ x = \sin y + C \]
where \(C = C_2 - C_1\) is a constant of integration.

### Step 5: Final solution
The final solution for the differential equation is:
\[ x = \sin y + C \]