Landau classical field theory question

[ASmc2]

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?

 

[ASmc2]

I assume it is as follows. He took du=dv in both frames, then said ds=1/gamma * cdt. But that still boters me.