### Abstract

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.

Original language | English (US) |
---|---|

Title of host publication | Mathematical time capsules |

Subtitle of host publication | Historicalmodules for the mathematics classroom |

Publisher | Cambridge University Press |

Pages | 223-228 |

Number of pages | 6 |

Volume | 9780883851876 |

DOIs | |

State | Published - Jan 1 2011 |

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*Mathematical time capsules: Historicalmodules for the mathematics classroom*(Vol. 9780883851876, pp. 223-228). Cambridge University Press. https://doi.org/10.5948/UPO9780883859841.030

**Newton's differential equation ẏ/ẋ = 1 − 3x+y+xx+xy.** / Kocak, Huseyin.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Mathematical time capsules: Historicalmodules for the mathematics classroom.*vol. 9780883851876, Cambridge University Press, pp. 223-228. https://doi.org/10.5948/UPO9780883859841.030

}

TY - CHAP

T1 - Newton's differential equation ẏ/ẋ = 1 − 3x+y+xx+xy

AU - Kocak, Huseyin

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84925059067&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925059067&partnerID=8YFLogxK

U2 - 10.5948/UPO9780883859841.030

DO - 10.5948/UPO9780883859841.030

M3 - Chapter

AN - SCOPUS:84925059067

VL - 9780883851876

SP - 223

EP - 228

BT - Mathematical time capsules

PB - Cambridge University Press

ER -