I'm getting my Hawkeye ready for some summer riding and when I went to start it the battery was dead. I jumped it and it started fine, but as soon as I disconnected the jumper cables it died. When I pull start it connected to the jumper cables it starts, but if I pull start it without the jumper cables connected it won't. I'm not that experienced with bikes, but every bike I've had before this was able to run with a bad battery as long as it was able to be started. What's the deal?

 

Could be the solenoid.