Question: Solve the DE (xsiny/x)dy=(ysiny/x -x)dx.
OK, I am reinterpreting/rewriting the above expression to give sense. Please use brackets to avoid confusion and ambiguity.
The DE is x.sin(y/x) dy = (y.sin(y/x) - x) dx, giving
dy/dx = y/x - 1/sin(y/x)
let v = y/x, then y = vx and dy/dx = v + x.dv/dx
Substituting for v and dy/dx into the DE,
v + x.dv/dx = v - 1/sin(v)
x.dv/dx = -1/sin(v)
-sin(v) dv = (1/x) dx
integrating both sides,
cos(v) = ln(x) + K
cos(y/x) = ln(x) + K
y = x.acos(ln(x) + K)