Frist-Order ODEs e^x(cosydx-sinydy)=0 ?

We can write this differential equation into differential form as:

# cosy-sinydy/dx=0 #

# :. dy/dx=cosy/siny #
# :. tanydy/dx = 1 #

Which is now a separable ODE, so "separating the variables" gives us:

# int \ tany \ dy = int \ dx #

And integrating we get

# ln | sec y| = x+C #

Which we can rearrange as:

# |sec y| = e^(x+A) #
# :. |1/cosy| = e^(x+A) #
# :. |cosy| = 1/e^(x+A) #
# :.

|cosy| = e^(-(x+A)) #

Putting #C=-A# this leads to the two solutions:

# cosy = +- e^(C-x) #
# :. y = +- arccos(e^(C-x)) #