How Bernoulli Solved Bernoullis Differential Equation

Adam E. ParkerWittenberg Universityaparker@wittenberg.eduhttp://userpages.wittenberg.edu/aparker/OriginalSourcesThe Benefits of Original Sources in an Ordinary Differential Equations (ODE) Class.

Thanks and short description of Wittenberg.1History of Using Historical DocumentsMotivated 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 & PengelleyThe ABCD of using history of mathematics in the (undergraduate) classroom by Siu Man-KeungWhat 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 OteroEtc. If Original Sources can motivate a Calculus or General Education course, why not my ODE class?Most of these talk of motivating students for which the mathematics itself doesnt do it. As such, they seem to concentrate on lower-level courses.2Why 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.

First should be taken with a grain of salt. Its possible that there are earlier orrucrances, but Ive found at least one citation of these being the first.Sources can be found on the handout.Please be patient with my translations. I dont speak latin, french, german, etc.3

Ex 1 Exact Differential EquationA 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!

The thing I want student to take away is that checking the hypotheses is important. While there may be more than one way to approach a problem, there are certainly wrong ways as well.4

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.

The thing to take away is that there are more than one way to approach a problem. And, mathematics really is an evolution of ideas, as we dont see the full variation of parameters till 1775.5Ex 3 The Vanishing WronskianThe 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.

Again, I want students to know to check the hypotheses. If Newton and the editor can make mistakes, so can they.6

Discover New MethodsEx 2 Linear 1st Oder D.E.A differential equation of the form

is called a linear first order differential equation.In 1694, Leibniz writes to lHopital that he can reduce it to quadratures by a change of variables.*In 1697, Bernoulli can solve it using variation of parameters as a special case of the Bernoulli Differential Equation.*In 1763, Euler solves this using integrating factors.*Today, we most closely use Eulers method.

Ex 4 Reduction of OrderGiven r solutions to an nth order differential equation, it is possible to reduce the order to n-r.Originally due to Lagrange, who told DAlembert, who responded that he had his own method. They each published in the same issue of Misc. Taurinensia in 1766.We teach DAlemberts method.

Students should understand the multiple approaches to a problem, as well as how mathematics is communicated.8Ex 5 Numerical MethodsRunge-Kutta Methods Numerically solve an ODERunge 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.

Again, many methods to solving a problem.9Used throughout the course.OverviewExistence / UniquenessSeparation of VariablesExact Differential EquationsLinear 1st orderReduction of OrderFundamental SystemsWronskiansHigher Order Linear w/ Constant Coeffs.Bernoulli D.E.Cauchy-Euler D.E.Variation of ParametersSeries SolutionsNumerical Methods(Missing Systems)

Excellent Source of Homework/ Exam Questions / ProjectsHomework 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/

BibilographyHandout with Citations of Original Sources.E. L. Ince Ordinary 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 1-20M. Krusemeyer, Why Does the Wronskian Work? The American Mathematical Monthly. Vol 95, No 1 (1988), pp. 46-49.ThanksDanny 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