Chapter 7 - Section 7.2 - Exponential Change and Separable Differential Equations - Exercises - Page 409: 16

$e^{-y}+e^{\sin x}=c$

Given: $\sec x\dfrac{dy}{dx}=e^{y+\sin x}$ Re-arrange the given equation and integrate as follows:. Then $\int (\cos x) e^{\sin x} dx= \int e^{-y} dy$....(1) Let us take the help of $u$ substitution. plug $u=\sin x \implies du =\cos x dx$ Equation (1) becomes: $ \int e^{u} du=-e^{-y}+c$ $\implies e^{u}=-e^{-y}+c$ Hence, $e^{-y}+e^{\sin x}=c$