Ordinary Differential Equations

Abstract

This chapter discusses ordinary differential equations. The most important mathematical model for physical phenomena is the differential equation. Motion of objects, fluids, and heat flow, bending and cracking of materials, vibrations, chemical reactions, and nuclear reactions are all modeled by differential equations. If a differential equation has one independent variable, then it is an ordinary differential equation. If the differential equation involves more than one independent variable, then it is a partial differential equation. Differential equations are often simplified or idealized models. A natural idea for numerical methods for differential equations is to replace the derivatives by differences and then solve the resulting difference equations. To be efficient, accurate difference approximations need to be used. This means that a first-order differential equation is replaced by a second-, third-, fourth-, or higher-order difference equation. A problem that immediately arises is that the differential equation has one solution, while a fourth-order difference equation has four solutions. Care must be made to get the right solution of the difference equation. This is not always easy or even possible; many methods that look good on the surface are, in fact, useless because one normally does not get the right solution. This difficulty permeates the study of numerical methods for differential equations and greatly complicates the theory and practice of solving differential equations.