This article refers to the 2007 model Honda Odyssey which features a 2.4 litre four cylinder engine. However this particular fault is very common for all Odysseys, the Accords (both V6 and the four cylinders) and CRVs.
The fault is a rapid oil leak from the power steering system. In many cases the high pressure line will be wet with oil but no visible leak to explain how this has happened. The leak is the high pressure hose itself. The rubber of the hose perishes over time and the oil slowly, or in some cases rather rapidly seeps through the hose.
Depending on the vehicle the hose itself is not too hard to remove. If you can identify more then three types of cars based off their emblems than you could change this hose. Two ten mil bolts on top of the pump, then either undo the hose at the rack or in newer models you can seperate the rubber one from the metal part at a swage fitting. A few swear words later, power steering oil running down your arm and into your arm pit and you’re half way to a qualified motor mechanic.
Replacement hoses. Genuine replacement hose can be very expensive. We always opt for a hose remanufacture. Companies such as ENZED or Quick fix can do this at a fraction of the cost of the new hose. Then slap it back together and you’re done… except for a few important items.
First replace the O ring where the hose bolts to the pump. Air condition O rings will do the trick.
Second make sure you use only genuine Honda power steering oil. Its cheap and using Dextron 3 (power steering oil most cars take) will damage the Honda system.
Bleed the system by going lock to lock several times with the cap off the power steering and the car off. This will help get air out of the system.