Prof. Matthew Westerhoff
There are numerous ways that can be used to imprecise solutions to a differential equation. One of the most basic explicit methods is Euler’s method. Euler’s method is the simplest for approximating solutions that do not change rapidly and for numerical integration of ODE. Also, Maple software is playing a significant role in learning DE to the students. On this project we will discuss how were the methods implemented using Maple software, what kind of problems we did encounter while working with the code, and the comparison of Euler’s solution with analytical solution.
Different methods were discussed in class but we are going to use one of the simplest explicit and easiest method to use here. This method was originally devised by Leonhard Euler and is called, Euler’s Method. Let’s start with a general first order Initial value problem. To apply Euler’s method to an IVP dy/dx = f (x, y), one need only change the initial line of the program, in which the function f is defined. Where f(t,y) is a known function and the values in the initial condition are also known numbers. To increase the number of steps one need only change the value of N specified in the second line of the program. Euler’s method is not an efficient numerical method, but many of the ideas involved in the numerical solution of differential equations are introduced most simply with it.
Using Euler’s method, we must first define the solution and give some numerical illustration. To apply Euler’s method to a differential equation dy/dx = f (x, y), one need only change the initial line of the program, in which the function f is defined. To increase the number of steps one need only change the value of N specified in the second line of the program. We illustrate below the implementation of Euler’s method in Maple software. We begin this project by implementing Euler’s method on Maple software. Test the code by application first to the IVP. I encountered some errors using the Maple when running the code, such as: (in dsolve) y(x) and y cannot both appear in the given ODE, unable to match delimiters (the expression was not complete). But I overcome these problems through maple online help, it gives you a link whenever you encounter some errors, it gives you the possible errors and solution on how to fix errors.
Often it is not possible or needed to solve a differential equation analytically, one turns to numerical or computational methods. Comparing Euler’s solution with the analytical solution there are some error percentage between the two. Analytical solution is exact, while Euler’s solution has some errors. Analytical solution can be derived by the separation of variables.
In summary, Euler’s method is the simplest numerical method for solving the initial value problem for approximating solutions and for the numerical integration for ODE. We illustrated the implementation of Euler’s method in Maple software, as well as the analytical solution. Comparing both Euler’s and analytical solution, we had some error percentage with the outcome. Despite the errors I’ve encountered, specifically while working with the code, I was able to overcome those problems with the help of Maple help links. Maple has a significant role in learning Differential Equation.