euler’s method – Numerical Methods Guy

Euler’s Method Example for FE Exam

“The Fundamentals of Engineering (FE) exam is generally the first step in the process of becoming a professional licensed engineer (P.E.). It is designed for recent graduates and students who are close to finishing an undergraduate engineering degree from an EAC/ABET-accredited program” – FE Exam NCEES

For most engineering majors, numerical methods is a required portion of the math part of the examination. Here is an example of using Euler’s method to numerically solve an ordinary differential equation.

This post is brought to you by

Holistic Numerical Methods Open Course Ware:
- Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com;
- Introduction to Matrix Algebra for the STEM undergraduate at http://ma.MathForCollege.com
the textbooks on
- Numerical Methods with Applications
- Introduction to Programming Concepts Using MATLAB
the Massive Open Online Course (MOOCs) available at

Global truncation error in Euler’s method

Illustrate through an example that the global truncation error in Euler’s method is proportional to the step size.

euler_truncation_global_pic1
euler_truncation_global_pic2

This post is brought to you by

Holistic Numerical Methods Open Course Ware:
- Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com;
- Introduction to Matrix Algebra for the STEM undergraduate at http://ma.MathForCollege.com
the textbooks on
- Numerical Methods with Applications
- Introduction to Programming Concepts Using MATLAB
the Massive Open Online Course (MOOCs) available at

Local truncation error is approximately proportional to square of step size in Euler’s method

Question: Show that the local truncation error in Euler’s method is proportional to the square of the step size.

euler_truncation_local_pic1 euler_truncation_local_pic2

This post is brought to you by

Holistic Numerical Methods Open Course Ware:
- Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com;
- Introduction to Matrix Algebra for the STEM undergraduate at http://ma.MathForCollege.com
the textbooks on
- Numerical Methods with Applications
- Introduction to Programming Concepts Using MATLAB
the Massive Open Online Course (MOOCs) available at

Example: Solving First Order Linear ODE by Integrating Factor

I have a audiovisual digital lecture on YouTube that shows the use of Euler’s method to solve a first order ordinary differential equation (ODE). To show the accuracy of Euler’s method, I compare the approximate answer to the exact answer. A YouTube viewer asked me: How did I get the exact answer?

In this blog, I use the integrating factor method to find the exact answer, because that is the method the viewer was using to solve the ODE exactly. So here it is and in two future blogs, I will show the same example being solved by 1) Laplace transforms and 2) the classical (complementary + particular) solution techniques.
Solving First Order Linear ODE by Integrating Factor

The pdf file of the solution is also available.

This post is brought to you by

Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com,
the textbook on Numerical Methods with Applications available from the lulu storefront,
the textbook on Introduction to Programming Concepts Using MATLAB, and
the YouTube video lectures available at http://nm.mathforcollege.com/videos

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