Chapter 8 - Mathematical Modeling With Differential Equations - 8.2 Separation Of Variables - Exercises Set 8.2 - Page 575: 7

$$y = \ln ( \sec x +c) $$

Given $$ e^{-y}\sin x -y'\cos^2x=0 $$ Then $$y'= e^{-y}\sec x\tan x$$ Separate variables \begin{align*} \int e^ydy &= \int \sec x\tan x dx \\ e^y&= \sec x +c\\ y&= \ln ( \sec x +c) \end{align*}