eulermethod3
Post on 13-Jul-2015
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What is Euler’s [OIL-er’s]
Method?• Numerical approach to approximating the
particular solution of the differential equation
– y’=F(x,y)
• TRANSLATION: Essentially, you take an initial
condition (a point), and use that condition as a
starting point to take a small step h along the
tangent line to get to a new point. Rinse and
repeat using the new point as a starting point.
• *Note: The smaller the steps used, the more
accurate your graph will be.
Explanation of Euler’s Method
• y’= F(x,y) w/ initial condition (x0,y0)
– x1= x0+h (h is your step)
– y1=y0+hF(x0,y0)
• Plug your initial condition into this formula
to find (x1,y1) and RINSE AND REPEAT
this is the slope of the tangent line at your initial point
F(x,y)
• How to write a first-order differential
equation in the form
– Example:
yxfdx
dy,
50,3.12 yeydx
dy x
is rewritten as
50,23.1 yyedx
dy x
so that
yeyxf x 23.1,
Example 1: page 411 #69
Use Euler’s Method to make a table of
values for the approximate solution of the
differential equation with the specified
initial value. Use n steps of size h.
y’=x+y y(0)=2 n=10
h=0.1
Example 1: page 411 #69
y1=y0+hF(x0,y0)
y1=2+(0.1)(0+2)
y1=2.2
y2=y1+hF(x1,y1)
y2=2.2+(0.1)(0.1+2.2)
y2=2.43
Y(0)=2N=10H=0.1
Example 1: page 411 #69
n 0 1 2 3 4 5 6
Xn 0 0.1 0.2 0.3 0.4 0.5 0.6
Yn 2 2.2 2.43 2.693 2.992 3.332 3.715
Real Life Application…
In 2000, the national debt of the United States was 5.6 trillion dollars and
in 2001, the debt rose to 5.8 trillion dollars. Using Euler’s method, one
could have estimated the national debt for the following years.
In year 2000 $5.6 trillion in
2001 $5.8 trillion is a 2%
increase.
t= years where t=0 is 2000, w=debt in billions of dollars
w’=.02w
Using step size h=1 for the range (0,10) the
first step would look something like…
T0=0 W0=5628.7
W1=W0+F(t0,w0)(h)
5628.7+(.02 x 5628.7)(1)
w(1)=$5741.3 in billions.
So 2001, the estimated value of the
national debt is about 5.7 trillion dollars.
Still pretty close to the actual value…
If you keep going…year debt w'=.02w
0 5628.7 112.5741 5741.274 114.82552 6889.529 137.79063 8267.435 165.34874 9920.921 198.41845 11905.11 238.10216 14286.13 285.72257 17143.35 342.8678 20572.02 411.44059 24686.43 493.7285
10 29623.71 592.4743
You eventually get the estimated value
for the annual debt of the United States
up to year 10, 2010.
However…, as you solve
for more values of w, it gets
more inaccurate because
Euler’s method is just an
estimation.
By looking at the actual values of
each year’s debt we can see the
inaccuracy of the Euler’s method
for this step size.
year debt w'=.02w0 5628.7 112.5741 5741.274 114.82552 6889.529 137.79063 8267.435 165.34874 9920.921 198.41845 11905.11 238.10216 14286.13 285.72257 17143.35 342.8678 20572.02 411.44059 24686.43 493.7285
10 29623.71 592.4743
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