Euler's Method: Answer to Example1

Example: Consider the autonomous differential equation with the initial condition




Hint: You may want to sketch the Slope Fields of this differential equation.

Find the first five terms of the Euler Approximation when tex2html_wrap_inline105 .
Is there a contradiction between the results of 1 and 2 ? If yes, explain what happened.


First, let us draw the slope fields of the differential equation.

Since tex2html_wrap_inline107 , any solution to the differential equation is increasing. This differential equation has one critical solution y=0. Since the initial condition satisfied by the solution to the IVP is y(0) = -0.1 < 0, then we have y(t) <0 for all t. We deduce from this that


Recall the formula which gives the euler's approximations to the solution


This gives the first five terms as:


According to the result in Part 1, the solution to the given IVP should always be negative and according to the above Euler's approximation the first term tex2html_wrap_inline123 is positive. This is our contradiction. As a matter of fact, according to the slope field, the Euler's approximation should continue to rise and even tend to tex2html_wrap_inline125, which even further contradicts the conclusion of 1.

The reason behind this is that tex2html_wrap_inline105 is a large step. So, after the first shot, it shoots above the critical solution y=0. Try to do the same problem with different step-sizes.

[Differential Equations] [Euler's Method]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2019 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour