6.1 day 2Euler’s Method

Leonhard Euler 1707 - 1783

Leonhard Euler made a huge number of contributions to mathematics, almost half after he was totally blind.

(When this portrait was made he had already lost most of the sight in his right eye.)

Greg Kelly, Hanford High School, Richland, Washington

Leonhard Euler 1707 - 1783

It was Euler who originated the following notations:

e (base of natural log)

f x (function notation)

(pi)

i 1

(summation)

y (finite change)

There are many differential equations that can not be solved.We can still find an approximate solution.

We will practice with an easy one that can be solved.

2dy xdx

Initial value: 0 1y

2dy xdx

0 1y

n nx nydydx dy 1ny

0.5dx

0 0 1 0 0 11 .5 1 1 .5 1.5

2 1 1.5 2 1 2.5

dy dx dydx

1n ny dy y

3 1.5 2.5 3 1.5 4.04 2.0 4.0

dy dx dydx

1n ny dy y

2dy xdx

n nx nydydx dy 1ny

0.5dx

0 0 1 0 0 11 .5 1 1 .5 1.5

2 1 1.5 2 1 2.5

0 1y

2dy xdx

0,1 0.5dx

2 dy x dx

2y x C

1 0 C

2 1y x

Exact Solution:

It is more accurate if a smaller value is used for dx.

This is called Euler’s Method.

It gets less accurate as you move away from the initial value.

The TI-89 has Euler’s Method built in.

Example: .001 100dy y ydx

0 10y

We will do the slopefield first:

6: DIFF EQUATIONSGraph…..

Y= 1 .001 1 100 1y y y We use:y1 for yt for x

MODE

6: DIFF EQUATIONSGraph…..

1 .001 1 100 1y y y

WINDOW t0=0tmax=150tstep=.2tplot=0xmin=0xmax=300xscl=10

ymin=0ymax=150yscl=10ncurves=0diftol=.001fldres=14

not criticalGRAPH

We use:y1 for yt for x

Y=

MODE

t0=0tmax=150tstep=.2tplot=0xmin=0xmax=300xscl=10

ymin=0ymax=150yscl=10ncurves=0diftol=.001fldres=14

WINDOW

GRAPH

While the calculator is still displaying the graph:

yi1=10

tstep = .2If tstep is larger the graph is faster.If tstep is smaller the graph is more accurate.

I Press and change Solution Method to EULER.

WINDOW

GRAPH

Y=

To plot another curve with a different initial value:

Either move the curser or enter the initial conditions when prompted.

F8

You can also investigate the curve by using .F3Trace

y1 2t

t0=0tmax=10tstep=.5tplot=0xmin=0xmax=10xscl=1

ymin=0ymax=5yscl=1ncurves=0Estep=1fldres=14

GRAPH

Now let’s use the calculator to reproduce our first graph:

2dy xdx

0 1y yi1 1

We use:y1 for yt for x

I Change Fields to FLDOFF.

WINDOW

Y=

Use to confirm that the points are the same as the ones we found by hand.

F3 Trace

TablePress

TblSetPress and set: tblstart... 0 tbl.... .5

This gives us a table of the points that we found in our first example.

TablePress

The book refers to an “Improved Euler’s Method”. We will not be using it, and you do not need to know it.

The calculator also contains a similar but more complicated (and more accurate) formula called the Runge-Kutta method.

You don’t need to know anything about it other than the fact that it is used more often in real life.

This is the RK solution method on your calculator.