Euler’s Method in Excel – Very Simple
October 19th, 2009 by SplineGuy
During Calculus last week, we covered slope fields and Euler’s method of first order initial-value problems of the form
![\begin{array}{rcl} \displaystyle \frac{dy}{dx} & = & f(x,y) \\[2ex] y(x_0) & = & y_0. \end{array}](/latexrender/pictures/e2d6721fc2360ffe9c73a4be3b57e8a2.gif "\begin{array}{rcl} \displaystyle \frac{dy}{dx} & = & f(x,y) \\[2ex] y(x_0) & = & y_0. \end{array}")
During class we demonstrated the use of slope field to perform a basic qualitative analysis. We utilized a couple of different online applets as well as Maple 13.
Visualizing Slope Fields and solutions
Another similar page, with zoom
Direction Fields – A Maple 13 Worksheet
Below is a short demonstration of how we set up a simple application of Euler’s Method in Excel 2007. Consider the simple initial value problem:
![\begin{array}{rcl} \displaystyle \frac{dy}{dx} & = & x(6-y) \\[2ex] y(0) & = & 0. \end{array}](/latexrender/pictures/7a3625a90510c457de39cfed09c24f86.gif "\begin{array}{rcl} \displaystyle \frac{dy}{dx} & = & x(6-y) \\[2ex] y(0) & = & 0. \end{array}")
Recall that Euler’s Method is given by
![\begin{array}{rcl} x_i &=& x_{i-1}+h\\[2ex] y_i &=& y_{i-1} + hf(x_{i-1},y_{i-1}) \end{array}](/latexrender/pictures/6d696cfe534656930445b1081824362f.gif "\begin{array}{rcl} x_i &=& x_{i-1}+h\\[2ex] y_i &=& y_{i-1} + hf(x_{i-1},y_{i-1}) \end{array}")
This is just a quick and dirty implementation of Euler’s method in Excel but it gets the job done.
At Wayland, there has been an increased emphasis on the need to "go green". I’ll admit to having inherited a very West Texas conservative viewpoint on the environment. I’ll even go so far as to admit not giving it much thought at all and just taking for granted the natural resources that are available to me.
Here’s something I already know: We are called to be stewards of many things as followers of Christ: stewards of our lives, stewards of our talents and gifts, stewards of our families, and stewards of our finances. God is creator and this world was deemed "good" by Him. We should take care of this Earth, a gift that He gave us to live on. The resources of this world are a gift and we are to be stewards of the Earth just as much as of our lives, family, finances and homes.
I love these kinds of puzzles because there is only one solution but there are several ways to get there. It’s at least as interesting to hear the different approaches as it is to solve it.
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with 4-digit number in which each of two digits appears two times. 
Clearly there are two lines of symmetry in the original problem so by reflection alone we come up with a total of four solutions: 
where 
and 
represent “flips” across the lines of symmetry and 
represents the identity, or the solution above. By 
, I mean the composition of the flipping operations or just consecutive flipping.
, but this is equivalent to 
