Differential compounded generators are used in Ward Lenard motor generator loops. The shunt fields on these generators are separately excited and when the shunt field polarity is reversed by the controller the series field helps drive the generator voltage to zero thus aiding in the reversal of current.
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A differential equation is a measure of change. If differencing with respect to time, it is the rate of change. Location, when differentiated, gives velocity. Velocity, when differentiated, gives acceleration. There are significant applications across all aspects of science.
All the optimization problems in Computer Science have a predecessor analogue in continuous domain and they are generally expressed in the form of either functional differential equation or partial differential equation. A classic example is the Hamiltonian Jacobi Bellman equation which is the precursor of Bellman Ford algorithm in CS.
If you are a scientist, engineer or mathematician, there are too many examples to list. If you aren't, then there are basically none, except in finance.