The car has to have a working (what they call) Auxiliary battery for the car to start and keep running. If it is a week battery that drains slowly over time, it should start with a jump and then continue running until you shut it off again (at least that is how my Prius works. BUT, if the battery has shorted cells, you can start it up with a jump but, like you said, when the cables are disconnected it would die again. Take your "Auxiliary" battery to an auto parts store and have then run it on their battery tester (it should be free of charge). They'll let you know what the deal is with the battery. Prius batteries have smaller posts than most so you should not just buy any battery from them unless the post size is the same as well as the physical size of the battery is the same. I usually just go to the Toyota dealer to get mine. You can buy just a regular battery that will fit in the space buy you'll have to convert your battery cables to fit the larger posts.