Step 1: Identify the Standard Form

The given equation is a first-order linear differential equation of the form:

dy/dx + P(x)y = Q(x)

Here,

P(x) = cot x

Q(x) = cosec x

Step 2: Find the Integrating Factor (IF)

Integrating Factor (IF) = e^{∫P(x) dx}

IF = e^{∫cot x dx}

∫cot x dx = ln(sin x)

IF = e^{ln(sin x)} = sin x

Step 3: Multiply the Equation by IF

sin x · dy/dx + y sin x cot x = sin x cosec x

Since:

sin x cot x = cos x

sin x cosec x = 1

The equation becomes:

d/dx (y sin x) = 1

Step 4: Integrate Both Sides

∫ d/dx (y sin x) dx = ∫ 1 dx

y sin x = x + c

Step 5: Solve for y

y = (x + c) / sin x

y = (c + x) cosec x