The general solution of e^x cosy dx – e^x siny dy = 0 is:

e^xcos y dx – e^xsin y dy = 0

⇒ e^xcos y dx = e^xsin y dy

Integrate

Substitute cosy = t hence which means siny dy = -dt

⇒ x = -log t + c

Resubstitute t

⇒ x = -log(cosy) + c

⇒ x + c = log(cosy)^-1

⇒ x + c = log(secy)

⇒ e^x+c = sec y

⇒ e^x × e^c = sec y

⇒ e^xcos y = e^-c

As e is a constant c is the integration constant hence e^-c is a constant and hence let it be denoted by k such that k = e^-c

⇒ e^xcos y = k