Rainbow, there's two issues in your message:
-growing dahlias hydroponically
-Slow growth of dahlia plant

Due to time limitations, I'll address the hydroponic question, and perhaps someone else will help with your other question.

You can start cuttings hydroponically, and if you have a big enough system with the correct levels of nutrients, it should even set buds and bloom... Let us know! But tuber production would be a challenge to support the weight over the water. Having the forming tubers under the water doesn't sound good to me, but I can't point to anything that says it won't work. You'd have to do your research with other tuber crop plants. Check out the below link, or google 'hydroponic tubers.'

I have started cuttings in soil, in a soil-less mix and in oasis, all with good success. Actually, I think the oasis was the easiest method that I will repeat this spring, as adding water to the bottom of the mini-containers was pretty easy to maintain (no nutrients added). The roots coming out of the oasis into the water was a clear indication of rooting success, too. I had no rotting of those feeder roots that wrapped themselves under the water level. My bet is that starting dahlia seed with hydroponics would work well, but I have no personal experience there.

Here is a link that might be useful: Gov research into tuber production with hydroponics

Hey thank you CCVacation! :)
your words really have a lot of experiences to learn.
So it needs more nutrient..What elements do you give it? NPK?
My dahlia is still a seedlings, and sadly it withered a few days ago, :(
honestly growing dahlias is a big mystery for me, for sometimes it likes water, but sometimes not.
But as a newbie I'll try and try...hope I'll succeed then to grow it hydroponically

Rainbow, where are you located?

There are tons of articles available that speak to the fertilizers recommended for dahlias, on dahlias.net and dahlia.org and elsewhere.

My theory is go as organic as possible, with low fertilizer numbers, because I don't want to burn the good bacteria and bugs/worms in/above my soil. I don't til, and haven't used noxious pesticides; many preying mantis, bees and worms are throughout my beds.

I use alfalfa pellets under my tubers, with lime and bonemeal, some three-month release fertilizer for tomatoes, and an organic low-number lawn fertilizer that takes all summer to break down. I dig in composted leaf mulch into the trenches in the fall so the worms will work on it during the winter. Overall, I encourage nature to keep the balance, instead of forcing it.

I have not grown dahlias from seeds, preferring to have established named cultivars that provide tubers for next season's plants.

Cheers,
CC