<s>[INST] Find the general solution of $(x + \sin{x} +\sin{y})\ \textrm{d}{x} + \cos{y}\ \textrm{d}{y}=0$ I have tried to make this equation seperable, homogeneous, exact and use integrating factors and nothing seems to be working. Could you tell me what form to use?
 [/INST] if you multply through by $e^x$ you have an exact diff eq.
$(x e^x + e^x \sin x + e^x \sin y)dx + (e^x \cos y) dy = 0\\
\nabla (x e^x-e^x + \frac 12 e^x \sin x - \frac 12 e^x \cos x + e^x \sin y) = (x e^x + e^x \sin x + e^x \sin y)dx + (e^x \cos y) dy\\ 
x e^x-e^x + \frac 12 e^x \sin x - \frac 12 e^x \cos x + e^x \sin y = C\\
-x + 1 - \frac 12 \sin x + \frac 12 \cos x +Ce^{-x} = \sin y\\
y = \arcsin( 1-x - \frac 12  \sin x + \frac 12 \cos x +Ce^{-x})$
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