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Applications of MATLAB : Ordinary Differential Equations ( ODE )

Textbooks on differential equations often give the impression that most differential equations can be solved in closed form, but experience does not bear this out. It remains true that solutions of the vast majority of first order initial value problems cannot be found by analytical means. Therefore, it is important to be able to approach the problem in other ways. Today there are numerous methods that produce numerical approximations to solutions of differential equations. Here, we introduce the oldest and simplest such method, originated by Euler about 1768. It is called the tangent line method or the Euler method. It uses a fixed step size h and generates the approximate solution. The purpose of this paper is to show the details of implementing a few steps of Euler’s method, as well as how to use built-in functions available in MATLAB (2005) [1]. In the first part, we use Euler methods to introduce the basic ideas associated with initial value problems (IVP). In the second part, we use the Runge-Kutta method presented together with the built-in MATLAB solver ODE45. The implementations that we develop in this paper are designed to build intuition and are the first step from textbook formula on ODE to production software.