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“But this will appear plainer by an Example or two.”
— Newton (1671)After outlining his general method for finding solutions of differential equations.
Introduction
In this note we redress Newton's solution to his differential equation in the title above in a contemporary setting. We resurrect Newton's algorithmic series method for developing solutions of differential equations term-by-term. We provide computer simulations of his solution and suggest further explorations.
The only requisite mathematical apparatus herein is the knowledge of integration of polynomials. Therefore, this note can be used in a calculus course or a first course on differential equations. Indeed, the author used the content of this paper while covering the method of series solutions in an elementary course in differential equations. Additional specific examples studied by the luminaries in the early history of differential equations are available in [1]. This work was supported by the National Science Foundation's Course, Curriculum, and Laboratory Improvement Program under grant DUE-0230612
Newton's differential equation
Newton's book [6], ANALYSIS Per Quantitatum, SERIES, FLUXIONES, AC DIFFERENTIAS: cum Enumeratione Linearum TERTII ORDINIS consists of one dozen problems. The second problem
PROB. II An Equation is being proposed, including the Fluxions of Quantities, to find the Relations of those Quantities to one another
is devoted to a general method of finding the solution of an initial-value problem for a scalar ordinary differential equation in terms of series. The equation in the title of the present paper (see also Figure 29.1) is the first significant example in the section on PROB. II.
Newton thought of mathematical quantities as being generated by a continuous motion.
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