What is the solution of \(y\,dx-x\,dy-y^{2}\cos x\,dx=0\)?
\(y=\dfrac{x}{c+\sin x}\)
Dividing the equation by \(y^{2}\) gives \(\dfrac{y\,dx-x\,dy}{y^{2}}=\cos x\,dx\), and the left side is exactly \(d\!\left(\dfrac{x}{y}\right)\). Integrating both sides gives \(\dfrac{x}{y}=\sin x+c\), so \(y=\dfrac{x}{c+\sin x}\).
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