I have a Vostro 1710 laptop that I recently aquired.  Here are the symptoms: 

1.  Battery charging light does not come on when adapter is plugged in.  Port where it plugs in feels loose.

2.  Laptop only starts if you remove and replace the battery (don't even need to push power button)

3.  Computer will boot to Safe Mode and run indefinitely (if adapter is plugged in)

4.  If you try and boot into normal Windows it will go through the boot sequence and about 5 seconds after you see the Windows logo it just shuts off

5.  Not sure about how well the battery holds a charge at this point

6.  If you remove the battery, plug in the adapter and press the power button nothing happens.

I am guessing that the charging port/harness will need to be replaced.   The fact that it will run in Safe Mode and not boot to Windows is confusing to me.  Any ideas about where to start looking is greatly appreciated.