7.Find an integrating factor and solve the given equation
$$
\underbrace{1}_{M}+(\underbrace{\frac{x}{y}-\sin(y)}_{N})y^{\prime}=0.
$$
$M_y=0$$N_x=\frac{1}{y}$
$\because$ $M_y\neq N_x$
$\therefore$ not exact
$R_1=\underbrace{M_y-N_x}_{M'}=\underbrace{0-\frac{1}{y}}_{N'}=-\frac{1}{y}$
$$\left. \begin{array} { l } { \mu = e ^ { - \int R_1 d r } = e ^ { - \int \frac { 1 } { y } } = e ^ { \operatorname { ln } | y | } = y } \\ { \therefore y + ( x - y \operatorname { sin } ( y ) ) y ^ { \prime } = 0 \quad \because M'_y = N'_x = 1 \quad \therefore \text{exact function}} \\{\therefore \exists\quad\varphi(x,y)\qquad s.t \,\varphi_x=M'=y}\\{ \psi = \varphi y d x = y x + h ( y ) }\\{ \varphi y = x + h ^ { \prime } ( y ) = x - y \operatorname { sin } ( y ) }\\{ h ^ { \prime } ( y ) = - y \operatorname { sin } ( y ) }\\{ h ( y ) = - \int y \operatorname { sin } ( y ) }\\{ = y \operatorname { cos } ( y ) - \operatorname { sin } ( y ) }\end{array} \right.$$
$$
\varphi=yx+y\cos(y)-\sin(y)=c
$$
