Landau classical field theory question

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ASmc2
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One page 24 of his book on classical field theory (4th edition), Landau derives the relativistic equation of motion for a uniformly accelarated particle. How does he get the differential equation that leads him to his result?
 
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I assume it is as follows. He took du=dv in both frames, then said ds=1/gamma * cdt. But that still boters me.