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Apple will not do that at all out of a bizarre principle and Apple also forbids AASPs do this kind of repair (the machine in question also is way out of warranty and AppleCare).
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If that is the case, then a good, and permanent, proper, fix would be to replace the cheap capacitor. The algorithm is based on a solution provided by user fabioroberto on MacRumors forums. MBPMid2010_GPUFix is an utility program that allows to fix MacBook Pro (15-inch, Mid 2010) intermittent black screen or loss of video. Performance will be a bit reduced by that. That can be made easier by using a ready made patch for that. That can be achieved by altering the kext AppleGraphicsPowerManagement.kext. If the capacitor is the problem, then one 'halfway' good solution is to circumvent the most problematic part of power management by lowering the power allowance for these dGPUs. But in reality it is most often only a tiny little capacitor that really has a flaw and interferes with power management of the dGPU. It is entirely possible that indeed the NVidia dGPU might be defective. It just appears like that because you notice the problems when the dGPU is 'switched on'. On most of these models it is not really the dGPU that is defective. While that circumvention is possible – either electrically, or with EFI/NVRAM hacking, or with moving the video driver kexts related to the NVidia-chip, it is by far not the best solution. Most owners of MBPs from 2010 with an apparently flawed NVidia dGPU should not seek to disable the GPU.
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