How to solve given ODE?

Question

$$\cos x\frac{dy}{dx}+y\sin x=\frac{\cos x}{e^x}$$

I have tried methods such as Integral multiplier , exact differential.The Integral of the solution part is difficult. $\displaystyle \int \sec x \frac{dx}{e^x}$

Can I solve it with another solution method?

Answer 1

Hint: Multiply both sides of ODE by $\sec^{2}(x)$, so you obtain a exact equation and if considerer the ODE as a linear equation, so you can take $\mu(x)=\sec(x)$ as integrate factor.

Answer 2

$$\cos x\frac{dy}{dx}+y\sin x=\frac{\cos x}{e^x}$$ $$\left (\frac{y}{\cos x}\right )'=\frac 1{\cos x e^x}$$ There is nothing you can do about the integral on RHS. The integral is hard to calculate. The mehod used to integrate the DE won't chane anything to this problem.Keep the integral in the solution. $$y(x)=C \cos x +\cos x\int \frac {dx}{\cos x e^x}$$