Need Help To Express The Voltage! Tonight!

This is from a FET-transistor.


https://i.imgur.com/wRvB9.jpg
**broken link removed**

I would like to express Uut(t) with Ugs!

I would write it as: Uut(t)= gmUgs*(RL//Rs) but then the voltage amplification is not reasonable.

Thanks for the help!
/Frida

what is on left side of the circuit (shows open)?
it looks like you are trying to solve small signal model for MOSFET.
show original circuit before reducing it to a small signal model.