Impact of Differential Equations on Transfer Functions, Impulse and Frequency Responses in Filters

For some examples of filters, defined by the "input-output" correspondence, a correspondence that can be translated in many cases by linear differential equations (first or second order filters), we specify the notions of step response, impulse response, transfer function, frequency response. We also specify the qualities of the filters considered from the point of view of causality, stability and rationality. To avoid any risk of confusion, some usual definitions and criteria are repeated below.