Does the resulting curve have to be perfectly mathematically correct on that particular surface? If now, you could create a curve on a sketch plane and use the Split Line function to project it onto the curved surface. That's how we modeled laser markings on 3D surfaces.
Your question would better fit in the Sketching section by the way, it has nothing to with SW drawings.
This is quite an involved process but may work
I used the edges of the unknown curve to create a sheetmetal body with a loft, I added some flats to one end so it can be unfolded, the equation curve was sketch on this flat face, I made a cut then refolded and converted into a 3d sketch
equationcurveonsurface.SLDPRT.zip 414.6 KB
I was able to get what I need, thanks to your method. Much appreciated.