Perhaps the differentials are transposed. The following is exact:
$$(\sin x - x\sin y) \,\mathrm d y + (\cos y + y \cos x)\,\mathrm d x = 0 \\ \implies y\sin x + x\cos y = c$$

Perhaps the differentials are transposed. The following is exact:
$$(\sin x - x\sin y) \,\mathrm d y + (\cos y + y \cos x)\,\mathrm d x = 0 \\ \implies y\sin x + x\cos y = c$$

Perhaps the differentials are transposed. The following is exact:
$$(\sin x - x\sin y) \,\mathrm d y + (\cos y + y \cos x)\,\mathrm d x = 0 \\ \implies y\sin x + x\cos y = c$$