Wright Math

In your previous work, you learned about how to "solve" a DE or an IVP using graphical methods, i.e., Slope Fields and Solution Curves.  In this next bit, you'll learn how to use Numerical Methods to approximate the solution to an IVP at a specific point.

The technique we'll be using is called "Euler's Method".  It basically just formalizes the process you went through in hand-sketching a Solution Curve through a Slope Field.

That process can be written as follows (you CS people might want to think about how you could write a little program to do this):

1.  Start at the IC point.

2.  Assess the slope at that point.

3.  Project forward (+x-direction) a little bit to get to the next point, using the slope from #2 as a guide for direction.

4.  Repeat the process  "Point - Slope - Project" until you've landed at the desired final x-value.

All we have to do is to write this process using the proper notation, organize our step-by-step movement is some comprehsible fashion, and voila!  We'll have a way to approximate new points on the solution curve.

Video of the Basic Idea

Tools:  Geogebra Slope Field with Euler's Polygonal Curve

Before reading the notes, have the Geogebra page linked above opened on a tab on your browser.

Notes:  Blank Copy      Filled In

Note:  Again, our book uses t rather than x, which isn't as intuitive since we think of the horizontal axis as x, not t, in general.  It's also not good prep for what happens in future courses, where both x and y are functions of the parameter t (as we saw in 12.1 [13.1]).  You can make the t's into x's in the homework problems then proceed.

 Assigned Problem