Find the particular solution of the differential equation $\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}-\frac{\mathrm{y}}{\mathrm{x}}+\operatorname{cosec}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)=0$; given that $\displaystyle \mathrm{y}=0$, when $\displaystyle \mathrm{x}=1$.

\[\begin{aligned} & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}-\operatorname{cosec} \frac{\mathrm{y}}{\mathrm{x}} \\ & \text { Put } \mathrm{y}=\mathrm{vx} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}} \\ & \Rightarrow \mathrm{v}+\mathrm{x} \frac{\mathrm{dv}}{\mathrm{dx}}=\mathrm{v}-\operatorname{cosec} \mathrm{v} \\ & \Rightarrow-\int \sin \mathrm{v} \mathrm{dv}=\int \frac{\mathrm{dx}}{\mathrm{x}} \\ & \Rightarrow \cos \mathrm{v}=\log |\mathrm{x}|+\mathrm{c} \\ & \Rightarrow \cos \frac{\mathrm{y}}{\mathrm{x}}=\log |\mathrm{x}|+\mathrm{c} \\ & \mathrm{x}=1, \mathrm{y}=\mathbf{0} \Rightarrow \mathrm{c}=1 \\ & \Rightarrow \cos \frac{\mathrm{y}}{\mathrm{x}}=\log |\mathrm{x}|+1 \end{aligned} \]