A graphical solution of a differential equation with application to hill’s treatment of nerve excitation

Linear differential equations with constant coefficients are very common in physical and chemical science, and of these, the simplest and most frequently met is the first-order equation ady /dt +y =f(t) , (1) wherea is a constant, andf(t) a single-valued function oft . The equation signifies that the quantityy is removed at a rate proportional to the amount present at each instant, and is simultaneously restored at a rate dependent only upon the instant in question. Familiar examples of this equation are the charging of a condenser, the course of a monomolecular reaction, the movement of a light body in a viscous medium, etc. The solution of this equation is easily shown to bey =e -t /a {y 0 = 1 / a ∫t 0 et /af(t) dt , (2) wherey 0 is the initial value ofy . In the case wheref(t) = 0, this reduces to the well-known exponential decay ofy .