I've fixed over a dozen D510 laptops with these symptoms, which turned out to be caused by a cracked solder joint for an inductor filter pair soldered to the board near the power jack (FL2). This component is a pair of inductors (or fuses or both, the "FL" abbreviation makes me think both) connected to the input power and ground.
The solder on the ground side of the component nearer the jack would often crack, resulting in a reading of about 6V on the output side of the filter, instead of the expected 19-20V. The broken solder joint resulted in too much resistance for it to work properly. Here are a couple examples where the broken solder joint is visible in the lower right portion of the picture:
By reflowing the solder and adding fresh solder to all four connections, the problem was immediately fixed and the connection strengthened. A fine tip aided in getting into the narrow space between the component and the power jack, so the repair could be performed without removing the board or jack.
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