$\displaystyle \phi(x,y)=e^{x}(x\cos y-y\sin y)=c , \ \ \ \ c \ \ \ constant \ \ $
what is the family of curves orthogonal to this phi.
This looks like a question from complex variable theory. If z = x + iy then $\displaystyle e^{x}(x\cos y-y\sin y)$ is the real part of the analytic function $\displaystyle ze^z$. For any analytic function f(z), the family of curves given by Im(f(z)) = const. will be orthogonal to the family of curves Re(f(z)) = const.