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Examples--Numerical
Methods
Euler's Method
Check out 10 steps of Euler's
Method applied to
y' = 1 - t + 4y y(0) = 1
EC: Find the analytical solution to the
equation above. The graph of the
analytical solution is given below.
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Example
Here is an example of the first step
(finding y1) in applying Euler's Method.
xo = 1, yo = e1/2,
h = 0.1
y1 = yo + (xoyo)(0.1)
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analytical solution is shown in
red above and the line
segment in
green above is
representing applying Euler's Method once to find y1.
Below is a graph of part of the
analytical solution of the differential equation above along with the tangent to the graph at the point where
x = 1.
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A Nice Euler's Method Example For The TI (The
Analytical Solution Is Shown Below)
The arrow above means press the STORE key.
The second ENTER
generates y1. Pressing ENTER again will produce y2.
Do you know why? Press the ENTER key repeatedly to generate
additional Euler solution values (y3, y4, . . .).
With yo = 10 you should get approximately y1 = 19, y2
= 34.39, y3 = 56.95, y4 = 81.47, y5 = 96.57, y6
= 99.88. The points along with the analytical solution are plotted in the
graph on the right below.
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To the left is the graph of the
solution to the logistics equation
dy/dx = .01y(100-y), y(0) = 10.
The solution is y = 100ex / (ex
+ 9). |
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| Euler's Method Extra Credit
Generate the data points for the example above
with x going from 0 to 250 and a step size of 2.5. The picture at
the right shows the analytical solution in red and the data points in blue
for x going from 0 to 40 with a step size of 2.5. Ideally you will
use a computer for this and generate a graph of the data points using your
computer and/or graphing calculator. What appears to be happening? |
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You could do this on your TI by entering the
following:
Keep pressing the enter key after the last
command to compute additional values for y.
Compare your results to this
excel sheet. pi/4 was rounded to
0.785398 |
Click on
the picture above to see the slope field added.
Try the First
Order DE Solution Grapher on this one and then try changing the derivative
to cos(xy).
Here is
an excel graph with h = 0.3927 (pi/8) and
h =
0.19635 (pi/16).
Maple Worksheet with graphs
Maple
Worksheet Maple
Worksheet Fields
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Here is an
audio/visual demonstrating implementing Euler's Method on a TI-84 with a
step size of pi/4 over an interval from 0 to 4pi. Click on the picture
at the right to see the result using a step size of pi/8 over an interval
from 0 to 2pi. The small boxes represent the Euler solutions. |
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| Another Euler's
Method Example Compared to an Analytical Solution |
Improved Euler's Method Compared To Euler's
Method--One Step
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Here is an Excel
graph comparing the Euler solution, Improved Euler solution and the
analytical solution to
y'(x) = cos(x),
y(0) = 0 over the interval from 0 to 2pi with a step size of
pi/8. |
Here is an animation
showing the first two steps of the Improved Euler Method applied to
y'(x) = 2cos(x), y(0) = 1
Quicktime Version
with a step size of pi/4 and comparing the
result to two steps of Euler's Method. |
Formulas for Applying the Improved Euler's Method and
a Classical Fourth Order Runge-Kutta Method
Maple Worksheet
comparing Runge-Kutta and Euler Methods
Here is a very nice Improved
Euler's Method applet by David Protas of California State University
that will both draw a graph of the Improved Euler solution and generate a
table of values.
Here is a nice java
applet demonstrating a variety of numerical methods.
Here is an Excel worksheet
that explores Euler's Method applied to dy/dx = sin(xy). In the data
sheets the first column corresponds to x-values and the other columns
correspond to y-values for various initial conditions. Here is a
Maple slope field picture with Runge-Kutta
solutions corresponding to initial conditions of y(0) = 1, y(0) = 1.5, and
y(0) = 2.
Another Excel worksheet,
this one exploring Euler's Method applied to dy/dx = (y^2)cos(x), y(0) =
1/2.
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