There is a hose in back of engine above transmission that has water dripping out

What is your question?

Hose is coming from the area of the middle of the car. If I look straight down the firewall behind the engine,center of car. The hose is short and doesn’t look like it was attached to anything. What is this hose. I was thinking condensation, but with this vehicle that should be running along passenger firewall.

What makes this problem better or worse?

Nothing

How long has your 2005 Toyota Corolla had this problem?

Just noticed it today