In this chapter we have consolidated the mathematical developments necessary to relate the physical system and the differential equation describing it. This included the derivation of the impulse response, step response, and convolution integral. These quantities, which are important to the theory of filtering, were shown to be a function of the L.I. solutions of the homogeneous equation. It is hoped that this time-domain approach has clarified the transition from the basic differential equation concepts to the everyday tools of the linear system analyst.