Solve the differential equation `(x dy-y dx)ysin(y/x)=(y dx+x dy)xcos(y/x)`.

Explanation

`(xcos"(y)/(x)+ysin"(y)/(x))ydx`
`=(ysin"(y)/(x)-xcos"(y)/(x))xdy`
`implies(dy)/(dx)=(y(xcos'(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)`
Let `y=vx`
`implies (dy)/(dx)=v+x(dv)/(dx)`
`impliesx(dv)/(dx)=(vcosv+v^(2)sinv)/(vsinv-cosv)-v`
`=(vcosv+v^(2)sinv-v^(2)sinv+vcosv)/(vsinv-cosv)`
`implies x(dv)/(dx)=(2vcosv)/(vsinv-cosv)`
`implies(vsinv-cosv)/(vcosv)dv=2(dx)/(x)`
`implies int(tanv-(1)/(v))dv=2int(dx)/(x)+logc`
`implies log secc-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`