A family of curves orthogonal to another family. How to graph this?

Question

I want to find a family of curves which are orthogonal to the family $$u(x,y) = e^{-x} \cos y + xy = const.$$

To this end, I check that $u$ satisfies Laplace's equation and so it has a harmonic conjugate $v$, which satisfies the Cauchy-Riemann equations, calculating $v$ (up to a constant, but that's irrelevant):

$$v(x,y) = -e^{-x} \sin y + \frac{1}{2}(y^2-x^2).$$

So the sought family is $$v(x,y)=const.$$

First of all, is this idea correct? Is this the only such family? Secondly, I want to graph this, but I don't have my Maple at the moment. Is Wolfram capable of doing this? I'd like to have a graph that shows that they are indeed orthogonal, so ideally a graph showing a few members of one family, say in red, and a few of the other, in blue.

Answer 1

Following the comment by (DeperHb). Yes, the idea is correct based on the fact

Geometrically, $u$ and $v$ are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function; the contours on which $u$ and $v$ are constant cross at right angles.

Here is a plot for $u(x,y)=2$ (in blue) and $v(x,y)=2$ (in green)