Find the solution of this integral

Question

I have to solve this differential equation:

After following the necessary steps, I have come to a point where I can't solve this integral:

Could you please help me?

Answer 1

One would say that $$ e^{\large \ln|\cos x|}=|\cos x| $$ where both sides exist, thus you have to evaluate $$ \int \frac{|\cos x|}{\cos x}\:dx. $$ Can you take it from here?

Answer 2

Your equation can be easy solved if we multiply both sides by $\cos x$, so we have: $$ y'\cos x-y\sin x=1 $$ and, noting that $-\sin x=\dfrac{d}{dx}\cos x$, we have:

$$ \dfrac{d}{dx}\left(y \cos x \right)=1 $$ i.e. $$y \cos x= x+c$$