$x\cfrac{{dy}}{{dx}} - y + x\sin \left( {\cfrac{y}{x}} \right) = 0$

: $x\cfrac{{dy}}{{dx}} - y + x\sin \left( {\cfrac{y}{x}} \right) = 0 \Rightarrow \cfrac{{dy}}{{dx}} = \cfrac{y}{x} - \sin \cfrac{y}{x}$

…(1)
Since R.H.S. is of the form $g(y/x)$ , so it is a homogeneous function of degree zero.

Therefore, equation (1) is a homogeneous differential equation.

To solve this, we put $y = vx$
$\Rightarrow \cfrac{{dy}}{{dx}} = v + x\cfrac{{dv}}{{dx}}$ , then (1) becomes $v + x\cfrac{{dv}}{{dx}} = v - \sin v$

$\Rightarrow {\rm{cosec}}vdv = - \cfrac{{dx}}{x}$

Integrating both sides,

we get
$\int {{\rm{cosec}}\,vdv} = - \int {\cfrac{{dx}}{x} + {C_1}}$

$\Rightarrow \log |\cos ecv - \cot v| = - \log |x| + {C_1}$

$\Rightarrow \log |(\cos {\rm{ec}}v - \cot v)|x = {C_1}$

$\Rightarrow |x({\rm{cosec}}v - \cot v)| = {e^{{C_1}}}$

$\Rightarrow x({\rm{cosec}}v - \cot v) = \pm {e^{{C_1}}} = C$

(say)
$\Rightarrow x\left( {{\rm{cosec}}\cfrac{y}{x} - \cot \cfrac{y}{x}} \right) = C$

which is the required general solution.