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Show that \( x y^{3}-x y^{3} \sin x=1 \) is an implicit solution to the differential equation \[ \frac{d y}{d x}=\frac{(x \cos (x)+\sin (x)-1) y}{3(x-x \sin x)} \] on the interval \( \left(0, \frac{\pi}{2}\right) \).

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