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Hello, The charging IC is6259HZ is burnt badly. Further the pins from 15 to 21  of is6259HZ  & R7001 & C7050 are also damaged. The battery was not getting charged with this. However, the green light is coming on the connector of Magsafe adapter. Further I have tried the other charged battery from my friend’s system in my system and it is working all fine. The computer is running all okay I guess with charged battery. Kindly suggest in detail (I can solder a few things): Is there any way to charge the battery externally using supply (via shorting any circuit etc.) ?Or, I could turn on the system with any power supply via battery ports?

The chip you are describing is U7000 (ISL6259HRTZ)

As you can see its the traffic cop of your power to your system! Sadly you’ll need to get your system to someone who has the deep skills to fix your logic board. This is beyond what a DIY can do! Even I have my limits as you likely have trace damage which makes this a much harder job. You only other option is finding a replacement logic board. Here’s one option 13” Unibody Macbook Pro Logic Board 2.3GHz i5 (Early 2011), Apple P/N 661-6078 Your system can support either the early or late 2011 boards as well as the 2012 with a slight adjustment to the displays LCVDS cable.