Green light cuts off when plugging into laptop

Remove the battery and try again.

If the laptop works ok with the charger connected (with battery taken out), then the battery needs to be replaced.

Refer to user guide on how to remove the battery.

YOU HAVE CHEAT

Start taking your laptop apart piece by piece and test it after each removed component:
1. Remove the hard drive and test the laptop.
2. Remove the DVD drive and test the laptop.
3. Remove the wireless card and test the laptop.
4. Unplug the display panel from the motherboard/video card and test the laptop.
etc....
Continue taking it apart until you have only three major components connected: motherboard (with video card), processor, and memory. If you still experience the same problem, most likely it's bad motherboard.
By the way, before you start, try reconnecting the memory module. If you have two modules installed, try removing them one by one and test the laptop with each one separately

Remove the battery and try again. If it solved the problem the battery is shorted and must be replaced. Look at the plug in the back of the laptop for the power. Is it lose? if it is lose you have a short from it touching contactd together and it must be replaced. If neither of these work try using a loner power supply to see if it is that. The D600 has lots of Power problems on of witch is the Motherboard itself. Hope this helps you out.

Power supply problem or power supply connector problem. I can take a look at it for you if you want.

The battery is dead and needs to be replaced. (It may have some power but not enough to boot the system.)

Hello,
Green light of the charger off means there is a short circuit, maybe exactly at the plug, so battery is impossible to charge. If not this, smth probably has fried.

Most likely the motherboard is bad. About $100 or so on Ebay

you'll need to replace the systemboard

It's definitely the motherboard based on the symptoms. You can send it to their out of warranty depot. Or you can purchase the motherboard from their spare parts sales dept, then look for a local tech who can install it for you.