Don't know if it's the right place (there's no gardening board), but maybe some /an/on has greener thumbs than I.
I was given a very large branch of a lovely jasmine shrub from a relative, and I've placed short sections of it (~25cm) in water, hoping they'd send out roots from between the segments so I can later plant them in soil; but (while still alive and green) they don't take out roots, as other segmented plants have for me.
I've very little experience, is there a way to encourage it to grow, like adding certain nutrients to the water? Or is this method not even feasible with jasmine? It worked for me with others.
Usually you're supposed to take cuttings in the spring or summer when the plant is actively growing. Sticking the cuttings in as well draining potting mix and keeping it moist is usually enough to get it rooted.
There's a gardening thread on /out/ that may be able to help you more than /an/.
I've not rooted a Jasmine ;) but some plants can take root in just water and others need other methods of propagation.
The usual method I take is get some potting mix, put it in a plastic pot, dip my cutting in rooting hormone after taking a fresh clean diagonal cut below a node. I remove most of the leaves and all of the flowers so the plant doesn't respire and dehydrate and wilt or put all of its effort in to the flower. I place the cutting in to the pot and potting mix and keep it moist, not soaking wet. I don't have it in a tray, I put each cutting inside a supermarket shopping bag and tie a loose knot at the top. This creates a humid environment and means the plant doesn't ever dry out and doesn't get too soaking wet ever. After about a month or so in a cool, well lit but out of sunlight area, it should take root