Given: $$x\frac{dy}{dx} = \cot y - \csc y \cos x, \ \ \lim_{x \to 0}y(x) = 0,$$ find $$y\left(\frac{\pi}{2}\right)$$
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$\begingroup$$$x\frac{dy}{dx} = \cot y - \csc y \cos x \implies \sin y \frac{dy}{dx}-\frac{\cos y}{x}=-\frac{\cos x}{x}$$Let $\cos y =v \implies -\sin y \frac{dy}{dx}=\frac{dv}{dx}$The ODE becomes linear:$$\frac{dv}{dx}+\frac{v}{x}=\frac{\cos x}{x}$$Its integrating factor is $I=e^{\ln x}=x$, then$$\cos y=\frac{1}{x}\int \cos x dx+ \frac{C}{x}$$$$x\cos y=\sin x+C \implies C=0$$So $$y(\pi/2)=\cos^{-1}(2/\pi).$$
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