solve each of the following differential equations subject t

solve each of the following differential equations subject to given conditions, if any

(csc y)dx+(sec^2 x)dy=0

We can write it as

csc y dx =- sec^2 x dy

dx/sec^2 x = - dy/cscy

=> cos^2 x .dx = -siny.dy

We can write it as

(cos2x + 1).dx/2 = -siny.dy (cos2x = 2cos^2 x - 1)

-siny.dy = (cos2x + 1).dx/2 =

Integrate both the sides, we will get

cosy = 1/2(sin2x/2 + x) +C1

cosy = sin2x/4 + x/2 + C1

Multiply each side by 4, we will get

4cosy = sin2x + 2x + 4C1

Let 4C1 = C

then final solution is

4cosy = sin2x + 2x + C

I have used separation of variables to solve this problem.

The answer given is:

4 cos y = 2x + sin 2x + C