Variational Methods for Engineers with Matlab

5Variational Methods for Differential Equations

Differential equations are a major tool for the analysis of real systems, used by scientists, engineers, economists and many people interested in quantitative or qualitative analysis. Nowadays, different areas of knowledge use differential equations in order to generate mathematical models.

The history of differential equations is subject to controversy, since a seminal dispute opposed Isaac Newton and Gottfried Leibniz concerning the origin of differential and integral calculus. In 1675, Leibniz wrote in an unpublished letter the equality

which may be considered as the resolution of a differential equation. In fact, the problem of the determination of the tangent to a given curve leads to differential equations and has been studied since ancient times, even if the main tools of differential calculus were introduced by Newton and Leibniz. Until these times, general concepts and analysis concerning differential equations were not available, limiting developments concerning differential equations. Since Archimedes, researchers worked on the determination of the tangent to curves. For instance, Gilles de Roberval developed an ingenious method, based on the motion of a particle along the curve. Roberval observed — analogously to Aristotle, but using more developed mathematical tools — that the direction of the velocity of such a particle ...