Solve `(dy)/(dx)=(s in)/(sin2y-xcosy)`

Explanation

We have `(dy)/(dx) = (siny+x)/(sin2y-xcosy)`
`therefore cosy(dy)/(dx) = (siny+x)/(2siny-x)`
Put `siny=vx`
`therefore cosy(dy)/(dx) = v+x(dv)/(dx)`
So, from (1), we have
`v+x(dv)/(dx) = (vx+x)/(2vx-x)`
`rArr x(dv)/(dx) = (v+1)/(2v-1)-v`
`rArr int(2v-1)/(-2v^(2)+2v+1)dv=int(dx)/(x)`
`rArr -1/2log_(e)(-2v^(2)+2v+1)log_(e)x+log_(e)C`
`rArr log_(e)1/sqrt(-2v^(2)+2v+1)=log_(e)Cx`
`rArr 1/sqrt(-2v^(2)+2v+1)=Cx`
`rArr 1/sqrt(-2(sin^(2)y)/(x^(2))+2(siny)/(x)+1) = Cx`
`rArr C_(1) = -2sin^(2)y+2xsiny+x^(2)`