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My personal experience with differential equations is from a long time ago. I wold be interested in someone showing me the details of Einstein's differential equation derivation of the Lorentz transformations in "On the Electrodynamics of Moving Bodies" - 1905. i understand what he does after he gets the differential equation but not how he gets it. He starts with time T as a function of coordinates in the stationary frame based on his method of synchronizing clocks;

1/2[T(0,0,0,t) + T(0,0-,0,t+x'/(c-v)+x'/(c+v))] = T(x',0,0,t+x'/(c-v))

He states "Hence if x' be chosen infintesimally small,

1/2(1/(c-v)+1/(c+v))dT/dt = dT/dX' + 1/(c-v)dT/dt

or

dT/dx' + v/(c^2-v^2)dT/dt = 0 "

Where the coordinates are (x,y,z,t) and the small letter "d" in the equations is the lower case Greek delta indicating partial derivatives.

Anybody?