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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