Show that the given differential equation is homogeneous and solve each of them.`x(dy)/(dx)-y+xsin(y/x)=0`

Explanation

`x(dy)/(dx)-y+xsin"(y)/(x)=0`
`implies (dy)/(dx)-(y)/(x)+sin"(y)/(x)=0`
Let `y=vx`
`implies (dy)/(dx)=v+x(dv)/(dx)`
`implies v+x(dv)/(dx)-v+sinv=0`
`implies x(dv)/(dx)+sinv=0`
`implies intcosecvdv+int(dx)/(x)=logc`
`implies log(cosec v-cotv)+logx=logc`
`implies x(cosecv-cotv)=c`
`impliesx(1-cosv)=csinv`
`implies x(1-cos"(y)/(x))=c sin"(y)/(x))`