What Are Differential Equations: A Review Of Curve Families

Abstract NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract Session 2265 WHAT ARE DIFFERENTIAL EQUATIONS: A Review of Curve Families Andrew Grossfield College of Aeronautics Abstract This paper is a review of curve families in light of their importance in a course on differential equations. Many texts depict curve families but do not treat them as an important mathematical concept that can greatly add to a student’s comprehension of differential equations and continuous mathematics in general. It is not an accident that the solution of an nth order differential equation is an n parameter curve family. It is not an accident that the solutions of linear differential equations are linear curve families. The forms, features and properties of curve families are discussed. Also to be discussed are the relations of one parameter curve families to surfaces in three dimensional space. What are Differential Equations? It is a good question. A student, desirous of answering the question, is not going to obtain a decent answer in most text books on Differential Equations. These texts assert “An equation with derivatives.” Obviously! These equations have derivatives in them. But why? Who cares? Why would anyone be interested? That is as uninformative as telling students that functions are ordered pairs. What is needed is explanation. Look at the simplest of the differential equations, those with just one derivative, the first order differential equations. Instead of focusing on the equation, focus on the solution, an equation with an independent variable, a dependent variable and a parameter. Solutions of these differential equations are equations in three variables. A student who has completed a calculus course is familiar with equations in two variables which he/she may interpret as curves in a two dimensional plane. Now the student is embarking on a study of something completely new, a study of equations in three variables. Faculty, of course, are familiar with the graphical representations of equations in three variables as surfaces and curve families, but these ideas may not be familiar to students. Perhaps there may be a pedagogical advantage to ensuring that students understand the relationships between equations in three variables and surfaces and curve families. There may also be pedagogical advantages to providing at this time a discussion of surfaces and providing a discussion of curve families. It is easy to show that beginning with the equation for curve family, F( x, y, pi, p2,... ,p,, ) = 0, in n parameters one can derive an nth order differential equation. However; it is only the simplest h