3.求微分方程y'=(y)/(x)+cot(y)/(x)的通解.

令 $ u = \frac{y}{x} $，则 $ y = xu $，代入原方程得 \[ u + x \frac{du}{dx} = u + \cot u \implies x \frac{du}{dx} = \cot u. \] 分离变量得 \[ \tan u \, du = \frac{dx}{x} \implies -\ln |\cos u| = \ln |x| + C. \] 化简得 \[ \cos u = \frac{C}{x} \implies \cos \frac{y}{x} = \frac{C}{x}. \] 或者等价表示为 \[ \boxed{x \cos \frac{y}{x} = C}. \]