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The Benefits of Original Sources in an Ordinary Differential Equations (ODE) Class.

The Benefits of Original Sources in an Ordinary Differential Equations (ODE) Class. Adam E. Parker Wittenberg University aparker@wittenberg.edu http://userpages.wittenberg.edu/aparker/OriginalSources. History of Using Historical Documents.

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The Benefits of Original Sources in an Ordinary Differential Equations (ODE) Class.

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  1. The Benefits of Original Sources in an Ordinary Differential Equations (ODE) Class. Adam E. Parker Wittenberg University aparker@wittenberg.edu http://userpages.wittenberg.edu/aparker/OriginalSources

  2. History of Using Historical Documents • Motivated by the Ohio MAA Short course in the summer of 2008 at Xavier University. • Using original sources in your teaching has pedagogical value. • “Recovering Motivation in Mathematics: Teaching with Original Sources” by Laubenbacher & Pengelley • “The ABCD of using history of mathematics in the (undergraduate) classroom” by Siu Man-Keung • “What the Knowledge of the History of Mathematics and of the Particular Subject Can Offer Us” by Miguel de Guzman. • “Calculus From An Historical Perspective: A Course For Humanities Students” by Daniel Otero • Etc. • If Original Sources can motivate a Calculus or General Education course, why not my ODE class?

  3. Why in an ODE Class? • Sources are concentrated in the writings of relatively few people • First published accounts of Separable, Homogenous, First Order Linear, Higher Order Linear with Constant Coefficients, Series Solutions, Variation of Parameters, Bernoulli, Cauchy-Euler and Exact can all be found in just Newton, Leibniz, Johann Bernoulli, and Euler. • Sources are in general readily available. • Language is NOT typically an issue, since the course is more computation than proof.

  4. Ex 1 – Exact Differential Equation • A differential equation of the form is called exact if Then there is an equation with • Euler first published this condition in 1763. • Newton was confused on this point way back in 1671!

  5. Ex 2 – Bernoulli Differential Eq. • A differential equation of the form is a Bernoulli differential equation. • Jacob Bernoulli proposed the problem in 1695 in Acta. • Leibniz in the March 1696 Acta. said he could solve it. • Johann Bernoulli gives two solutions in March 1697. • Variation of Parameters in its generality is due to Lagrange in 1775.

  6. Ex 3 – The Vanishing Wronskian • The vanishing of the Wronskian is a necessary condition for linear independence, but not sufficient. • First noted by Peano in 1889, when he published two small articles in the same issue of Mathesis. • The sufficient conditions are scattered through other papers by Bocher and others.

  7. Ex 4 – Reduction of Order • Given r solutions to an nth order differential equation, it is possible to reduce the order to n-r. • Originally due to Lagrange, who told D’Alembert, who responded that he had his own method. • They each published in the same issue of Misc. Taurinensia in 1766. • We teach D’Alembert’s method.

  8. Ex 5 – Numerical Methods • Runge-Kutta Methods – Numerically solve an ODE • Runge gave the general method in 1895. • Kutta in 1901 the coefficients for the famous RK4. • In that same paper, Kutta has many other methods as well.

  9. Used throughout the course. • Overview • Existence / Uniqueness • Separation of Variables • Exact Differential Equations • Linear 1st order • Reduction of Order • Fundamental Systems • Wronskians • Higher Order Linear w/ Constant Coeffs. • Bernoulli D.E. • Cauchy-Euler D.E. • Variation of Parameters • Series Solutions • Numerical Methods • (Missing Systems)

  10. Excellent Source of Homework/ Exam Questions / Projects • Homework Example: • Exam Example: • Project Example: Compare the convergence of several of the numerical methods found in Kutta. • We have recently instituted a CLAC (Culture and Languages Across the Curriculum) at Wittenberg. http://www4.wittenberg.edu/features/language/

  11. Bibilography • Handout with Citations of Original Sources. • E. L. InceOrdinary Differential Equations Dover 1956. is an excellent overview. • Specific Sources: • S.S. Demidov, “On the history of the theory of linear differential equations” Archive for History of Exact Sciences. Vol 28, No 4 (1983). Pp 369-387. • J.C. Butcher, “A history of Runge-Kutta methods” Applied Numerical Mathematics. 20 (1996) pp 247-260. • P. Pragacz, “Notes on the life and work of Jozef Maria Hoene-Wronski” Trends in Mathematics – Algebraic Cycles, Sheaves, Shtukas, and Moduli. Birkhauser (2008) pp 1-20 • M. Krusemeyer, “Why Does the Wronskian Work?” The American Mathematical Monthly. Vol 95, No 1 (1988), pp. 46-49.

  12. Thanks • Danny Otero and David Pengelley for running the Ohio MAA short course. • Alisa Mizikar in our library for helping track down a few of the sources. • Dr. Brian Shelburne for his helpful suggestions on this talk. Questions? • Adam Parker at Wittenberg University: http://userpages.wittenberg.edu/aparker/OriginalSources

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