\begin{equation}
1+\left(\frac{x}{y}-\sin (y)\right) y^{\prime}=0
\end{equation}
\begin{equation}
M(x, y)=1 \quad \text { and } \quad N(x, y)=\left(\frac{x}{y}-\sin (y)\right)
\end{equation}
\begin{equation}
\frac{\partial}{\partial y} M(x, y)=0 \quad \text { and } \quad \frac{\partial}{\partial x} N(x, y)=\frac{1}{y}
\end{equation}
\begin{equation}
\frac{N_{x}-M_{y}}{M}=\frac{1}{y}
\end{equation}
\begin{equation}
\frac{d \mu}{d y}=\frac{N_{x}-M_{y}}{M} \mu=\frac{\mu}{y} \quad \Rightarrow \quad \mu=y
\end{equation}
\begin{equation}
y+(x-y \sin (y)) y^{\prime}=0
\end{equation}
\begin{equation}
\frac{\partial}{\partial y}(y)=1=\frac{\partial}{\partial x}(x-y \sin (y))
\end{equation}
\begin{equation}
\begin{array}{l}{\Psi_{x}(x, y)=y} \\ {\Psi_{y}(x, y)=x-y \sin (y)}\end{array}
\end{equation}
