`{xcos((y)/(x))+y sin((y)/(x))}y dx={ysin((y)/(x))-xcos((y)/(x))}x dy`

Explanation

`(xcos.(y)/(x)+y sin.(y)/(x))y dx`
`=(ysin.(y)/(x)-xcos.(y)/(x))x dy`
`implies (dy)/(dx)=(y(cos.(y)/(x)+ysin.(y)/(x)))/(x(ysin.(y)/(x)-xcos.(y)/(x)))`
`implies v+x(dv)/(dx)=(vx(xcosv+vxsinv))/(x(vxsinv-xcosv))`
`=(v(cosv+vsinv))/(vsinv-cosv)`
माना `y = vx implies (dy)/(dx)=v+x(dv)/(dx)`
`implies x(dv)/(dx)=(vcosv+v^(2)sinv)/(vsinv-cosv)-v`
`=(vcosv+v^(2)sinv-v^(2)sinv+vcosv)/(vsinv-cosv)`
`impliesx(dv)/(dx)=(2v cosv)/(vsinv-cosv)`
`implies (vsinv-cosv)/(vcosv)dv=2(dx)/(x)`
`implies int(tanv-(1)/(v))dv=2int(dx)/(x)+logc`
`implies logsecv-logv=2logx+logc`
`implies log((secv)/(v))=log(cx^(2))`
`implies (secv)/(v)=cx^(2)`
`implies secv=cx^(2)v`
`implies sec.(y)/(x)=cxy`