TITLE: To solve $y'=\frac{\cos y+y\cos x}{x\sin y-\sin x}$
QUESTION [6 upvotes]: $$y'(x)=\frac{\cos (y(x))+y(x) \cos (x)}{x \sin (y(x))-\sin (x)}$$
I am self-learning the differential equation from a textbook and I need some help with above equation. I learned I-factor (for 1st order linear ODE), Bernoulli's, and Riccati's. So the problem should be solved by them. The book doesn't have any examples using trig functions.
I tried wolfram alpha but the website could not give me "step-by-step". Does anyone know why? Wolfram Alpha never had this problem where it could not give "step-by-step".

REPLY [6 votes]: $$y'(x)=\frac{\cos (y(x))+y(x) \cos (x)}{x \sin (y(x))-\sin (x)}$$
we can re-write as 
$$
x \sin (y(x))y'-\sin (x)y' = -x\frac{d}{dx}\cos (y(x)) -\sin(x)y'=\cos (y(x))+y(x) \cos (x)
$$
collecting similar terms(by inspection that is)
$$
-x\frac{d}{dx}\cos (y(x))-\cos(y(x)) = -\frac{d}{dx}x\cos(y(x))
= \sin(x)y' + y(x)\cos(x) = \frac{d}{dx}y\sin(x)
$$
or
$$
-\frac{d}{dx}x\cos(y(x)) = \frac{d}{dx}y\sin(x)
$$
proceed :)