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6.1 day 2

Euler’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 ortrait was

made he had already lost

most of the sight in his right

eye.!

"reg #elly, $anford $igh %chool, &ichland, Washington

→

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  Leonhard Euler 1707 - 1783

't was Euler who originated

the following notations

e (base of natural log!

( ) f x (function notation!

π   (i!

i   ( )1−

(summation!∑

 y∆

(finite change! →

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)here are many differential e*uations that can not be sol+ed.

We can still find an aroimate solution.

We will ractice with an easy one that can be sol+ed.

2dy

 xdx

= 'nitial +alue0 1 y   =

→

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2dy

 xdx

=0 1 y   =

n   n x n y dydx   dy   1n y +

0.5dx  =

0   0 1   0 0 1

1 .5 1 1   .5 1.52   1 1.5 2   1   2.5

dydx dy

dx

⋅ =1n n y dy y ++ =

→

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3 1.5   2.5 3 1.5   4.0

4 2.0   4.0

dydx dy

dx

⋅ =1n n y dy y ++ =

2dy

 xdx

=

n   n x n y dydx   dy   1n y +

0.5dx  =

0   0 1   0 0 1

1 .5 1 1   .5 1.52   1 1.5 2   1   2.5

0 1 y   =

→

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2dy

 xdx

=   ( )0,1 0.5dx  =

2dy x dx=

2 y x C = +

1 0   C = +

21 y x= +

Eact %olution

→

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't is more accurate if a smaller

+alue is used for dx.

)his is called Euler’s Method.

't gets less accurate as you

mo+e away from the initial

+alue.

→

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)he )'-/ has Euler’s Method built in.

Eamle   ( ).001 100dy

 y ydx = −0 10 y   =

We will do the sloefield first

6 0' E34)'5%"rah7..

89   ( )1 .001 1 100 1 y y y′ = ∗ ∗ −We use

 y1 for  y

t  for  x

→

MODE

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6 0' E34)'5%"rah7..

( )1 .001 1 100 1 y y y′ = ∗ ∗ −

W'05W t:9:

tma91;:tste9.2

tlot9:

min9:

ma9

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t:9:

tma91;:

tste9.2

tlot9:min9:

ma9

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While the calculator is still dislaying the grah

yi1=10

tstep = .2

'f tste is larger the grah

is faster.'f tste is smaller the

grah is more accurate.

→

 ' >ress and change %olution Method to EULER.

W'05W

"&4>$

89

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)o lot another cur+e with a different initial +alue

Either mo+e the curser or enter the initial conditions

when romted.

→

F8

8ou can also in+estigate the cur+e by using .F3

)race

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y1 2t ′ =

t:9:

tma91:tste9.;

tlot9:

min9:

ma91:

scl91

ymin9:

yma9;yscl91

ncur+es9:

Este91

fldres91=

"&4>$

ow let’s use the calculator to reroduce our first grah

2dy

 x

dx

= 0 1 y   =

yi1 1=

We use

 y1 for  y

t  for  x

 '  ?hange ields to FLDOFF.→

W'05W

89

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  →

3se to confirm that the oints are the same as

the ones we found by hand.

F3 )race

)able>ress

)bl%et>ress and set tblstart... 0 tbl.... .5∆

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)his gi+es us a table of the oints that we found in our

first eamle.

→

)able>ress

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)he boo@ refers to an A'mro+ed Euler’s MethodB. We will

not be using it, and you do not need to @now it.

)he calculator also contains a similar but more comlicated

(and more accurate! formula called the &unge-#utta

method.

8ou don’t need to @now anything about it other than the fact

that it is used more often in real life.

)his is the solution method on your calculator.

π