website designing comapny in delhi ncr
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8- 1
Chapter 8 Differential Equations
• An equation that defines a relationship between anunknown function and one or more of its derivatives
is referred to as a differential equation.
• A first order differential equation:
• Example:
)! y x f dx
dy=
"#$"#%obtain we&and%n'(ubstituti
%
"'etweit(olvin'
#&at%condition boundarwith"
%
%
−===
+=
===
x y x y
c x y
x y xdx
dy
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8- 2
• Example:
• A second*order differential equation:
• Example:
)!%
%
dx
dy y x f
dx
yd =
)! x ycdx
dy−=
+%++ y xy x y ++=
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8- 3
,alor (eries Expansion
• -undamental case the first*order ordinar differential
equation:
.nte'rate both sides
• ,he solution based on ,alor series expansion:
$$ atsub/ect to )! x x y y x f
dx
dy===
∫ ∫ = x
x
y
ydx x f dy
$$
)! ∫ +== x
xdx x f y x g y
$
)!)!or $
)!)!+ and )!where
###)!++0%
)!)!+)!)!)!
$$$$
$
%
$$$
x f x g x g y
x g x x
x g x x x g x g y
==
+−
+−+==
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8- 4
Example : First-order
Differential Equation
1iven the followin' differential equation:
,he hi'her*order derivatives:
&at&such that2 % === x y xdx
dy
3nfor$
4
4
2
2
%
%
≥=
=
=
n
n
dx
yd
dx yd
xdx
yd
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8- 5
,he final solution:
& where
)&!)&!2)&!2&
)4!02
)&!)4!
0%
)&!)2)!&!&
02
)&!
0%
)&!)&!&)!
$
2%
2
$
%%
$
2
22
%
%%
=
−+−+−+=
−+
−+−+=
−
+
−
+−+=
x
x x x
x x
x x x
dx
yd x
dx
yd x
dx
dy x x g
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8- 6
x 5ne ,erm ,wo ,erms ,hree ,erms -our ,erms
& & & & &
&#& &   &
&#% &  $#6% %8
 &  %#&6 %#&76
 & %#% %#48 %#633
&#" & %#" 2#%" 2#26"
 & %#8 2#88 3#$74
 & 2#& 3#"6 3#7&2
 & 2#3 "#2% "#82%
 & 2#6 4#&2 4#8"7
% & 3 6 8
,able: ,alor (eries (olution
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8- 7
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8- 8
1eneral Case
• ,he 'eneral form of the first*order ordinar
differential equation:
• ,he solution based on ,alor series expansion:
$$
atsub/ect to )! x x y y y x f dx
dy===
###)!++0%
)!
)!+)!)!)! $$
$
$$$$$ +
−
+−+== y x g
x x
y x g x x y x g x g y
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8- 9
Eulers 9ethod
• 5nl the term with the first derivative is used:
• ,his method is sometimes referred to as the one-step
Euler’s method since it is performed one step at a
time#
edx
dy x x x g x g +−+= )!)!)! $$
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8- 10
Example: 5ne*step Eulers 9ethod
• Consider the differential equation:
• -or x &#&
,herefore at x&#& y!&22 !true value)#
&at&such that3 % === x y xdx
dy
∫ ∫ =&#&
&
%
&3 dx xdy
y
33&22#$2
3&
&#&
&2
==− x y
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8- 11
$#$$8262#iserror,he
 %74'!&#&$)
valueestimatedthestepsfiveafter$#$%of si;estepa-or
$#$%$822#toreducediserror,he
3%$"#&<)$"#&!3)=$"#&&$#&!)$"#&!)&$#&!
%#&%#$&<)&!3)=$$#&$"#&!)&!)$"#&!
:)$"#& and &!at
twiceequationsr+appl Euleand$#$"of si;estepa>se
value)#absolute!in $#$3&22error ,he
3#&<)&!3=&#$&)&#&!
'et we&#$)!of si;estepa?ith
%
%
%
$
=
=−+=
=+=−+=
==
=
=+=
=−=∆
g g
g g
x x
g
x x x
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8- 12
Errors with Euler@s 9ethod
• Local error : over one step si;e#
Global error : cumulative over the ran'e of the solution#
• ,he error ε usin' Euler@s method can be approximated usin' the
second term of the ,alor series expansion as
• .f the ran'e is divided into n increments then the error at the end
of ran'e for x would be nε .
<#=inmaximumtheis where
0%
)!
$%
%
%
%%
$
x xdx
yd
dx
yd x x −=ε
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8- 13
Example: Analsis of Errors
$$88#$)$%#$)!&#&)!3!"
$%%#$)$"#$)!&#&)!3!%
$33#$)&#$)!&#&!3 :&#&at
error theonlimitsupperthe$#$%#and$#$"$#&of si;esstep-or
)!3)8!0%
)! b boundediserrorthe,hus
8
&at&thatsuch 3
%
$%#$
%
$"#$
%
&#$
%
$
%
$
%
%
%
==
==
===
−=−
=
=
===
ε
ε
ε
ε
x
x x x x x x
xdx
yd
x y xdx
dy
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8- 14
,able: ocal and 1lobal Errors with a (tep (i;e of $#&#
x Exactsolution
Bumerical(olution
ocalError!)
1lobalError!)
& & & $ $
&#& !&2222  *%#8466&"& *%#8466&"&
&#% L$4446 ͳ *%#2$$3$4 *3#2768237
 %#"74 %#34 *$$2"7" *"#%288%7
 2#2%"2222 2#&24 *$2837% *"#4724438
&#" 3#&444446 2#7% *Ē *"*7%
 "#&%8 3#8% *&#&74$368 *4#$$4%3$%
 4#%&62222 "#833 *&#$"$8%"4 *4#$$36&8
 6#33%4446 6 *$#72&"4"6 *"#736487
 8#8&% 8#%74 *$#82%&78" *"#8""4"&3
% &$#222222 7#63 *$#638286& *"#63&72""
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8- 15
x Exactsolution
Bumerical(olution
ocalError!)
1lobalError!)
& & & $ $
&#$" &#%&$&446 &#% *$#83$&$36 *$#83$&$36
&#& !&2222 %$" *$#63$$""" *!"3%$7
&#&" Ǚ" ,%" *$#4"87738 *𕬂%6
&#% L$4446 %6 *$#"7%$&4% *%#%&"82%%
&#%" %#%6$8222 %#%&" *$#"2"6678 *%#3"86&"4
 %#"74 %#"%6" *$#38672$& *%#4284637
" %#736&446 %#84"" *$#3346"48 *%#66&$%2
 2#2%"2222 2#%2 *$#3&$7843 *%#84488$"
" 2#62&" 2#4%% *$#2674"$6 *%#7233648
&#" 3#&444446 3#3$%" *$#2"% *%#78
,able: ocal and 1lobal Errors with a (tep (i;e of $#$"#
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8- 17
x Exact solution Bumerical (olution ocal Error!) 1lobal Error!)
& & & $ $
&#$% &#$8&4&$6 &#$8 *$#&387&26 *$#&387&26
&#$3 &#&4438"2 &#&42%2% *$#&3$8%&7 *$#%687$$"
&#$4 &#%"3488 &#%3764 *$#&2236%8 *$#27%646
&#$8 ê%8%6 𷡮 *$#&%46488 *$#37%8&28
&#& !&2222  %74 *$#&%$4%7 *$#"8$7324&#% L$4446 "2&% *$#$742343 *$#87$27%3
 %#"74 %#"4838 *$#$672$&" *&#$4$$7%3
 2#2%"2222 2#%86$3 *$#$446%$& *&#&"&"428
&#" 3#&444446 3#&&48 *$#$"6$88 *&#&748
 "#&%8 "#$4864 *$#$37"$4 *&#%&26%8"
 4#%&62222 4#&3&7% *$#$323$"" *&#%&%7"2
 6#33%4446 6#2"2%8 *$#$283$7% *&#%$&$$2%
 8#8&% 8#6$683 *$#$23%"42 *&#&8%$%3"
% &$#222222 &$#%&24 *$#$2$64&2 *&#&"86$76
,able: ocal and 1lobal Errors with a (tep (i;e of $#$%#
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8- 21
(econd*order un'e*utta 9ethods
• ,he modified Eulers method is a case of the second*
order un'e*utta methods# .t can be expressed as
xh x x x
x x g y x g y
h y xhf yh x f y x f y y
ii
iiii
iiiiiiii
∆=∆+=∆+==
++++=
+
+
+
)! )! where
))<!!)!="#$
&
&
&
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8- 22
• ,he computations accordin' to Eulers method:
&# Evaluate the slope at the start of an interval that is
at ! xi yi) #
%# Evaluate the slope at the end of the interval
! xi+&
yi+&
) :
2# Evaluate yi+& usin' the avera'e slope S & of and S % :
)!& ii y x f S =
)! &% hS yh x f S ii ++=
hS S y y ii )!"#$ %&& ++=+
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8- 24
• ,he computational steps for the third*order method:
&# Evaluate the slope at ! xi yi)#
%# Evaluate a second slope S % estimate at the mid*point
in of the step as
2# Evaluate a third slope S 2 as
3# Estimate the quantit of interest yi+& as
)"#$"#$! &% hS yh x f S ii ++=
)%! %&2 hS hS yh x f S ii +−+=
hS S S y y ii <3=4
&2%&& +++=+
)!& ii y x f S =
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8- 25
-ourth*order un'e*utta 9ethods
&# Compute the slope S & at ! xi yi)#
%# Estimate y at the mid*point of the interval#
2# Estimate the slope S % at mid*interval#
3# evise the estimate of y at mid*interval
# atthatsuch )! $$ h x x x y y y x f dx
dy
=∆===
)!& ii y x f S =
)!%
%F& iiii y x f h
y y +=+
)"#$"#$! &% hS yh x f S ii ++=
%%F&
%
S h
y y ii +=+
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8- 26
"# Compute a revised estimate of the slope S 2 at mid*
interval#
4# Estimate y at the end of the interval#
6# Estimate the slope S 3 at the end of the interval
8# Estimate yi+& a'ain#
)"#$"#$! %2 hS yh x f S ii ++=
2& hS y y ii +=+
)! 23 hS yh x f S ii ++=
)%%!4
32%&& S S S S h y y ii ++++=+
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8- 27
Gredictor*Corrector 9ethods
• >nless the step si;es are small Eulers method
and un'e*utta ma not ield precise
solutions#
• ,he Gredictor*Corrector 9ethods iterate
several times over the same interval until the
solution conver'es to within an acceptable
tolerance#
• ,wo parts: predictor part and corrector part #
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8- 28
Euler*trape;oidal 9ethod
• Eulers method is the predictor al'orithm#
• ,he trape;oidal rule is the corrector equation#
• Eluer formula !predictor):
• ,rape;oidal rule !corrector):
,he corrector equation can be applied as man times as
necessar to 'et conver'ence#
H
H&
i
i jidxdyh y y +=+
<=
% &&H
H&
−+
+ ++= jii
i ji
dx
dy
dx
dyh y y
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8- 29
Example 8*4: Euler*trape;oidal 9ehtod
&at&thatsuch :Groblem === x y y xdx
dy
&#&)&!&#$&
&#$
&&&
is &#&atforestimate)!predictor initial,he
$&
$$
H$$&
$$
=+=
+=
==
=
y
dx
dy y y
dx
dy
x y
&"247#&&#&&#&
:estimatetheimprovetousedisequationcorrector,he
$&
==dx
dy
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8- 30
[ ]
[ ]
[ ] &$687#&&"68%#&&
%
&#$&
%
&"68%#&&$687#&&#&
&$687#&&"66&#&&%
&#$&
%
&"66&#&&$648#&&#&
&$648#&&"247#&&%
&#$&
%
%&$$
H$2&
%&
&&$$
H$%&
&&
$&$$
H$&&
=++=
++=
==
=++=
++=
==
=++=
++=
dx
dy
dx
dyh y y
dx
dy
dx
dy
dx
dyh y y
dxdy
dx
dy
dx
dyh y y
#haveweAnd
#&#&at&#&$687toconver'es (ince
2&H&
%&2&
y y
x y y y
=
==
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8- 31
%%246#&)&"68%#&!&#$&$687#&
:equation predictorthe%#&atof estimate -or the
H&H&$% =+=+=
=
dx
dyh y y
x y
[ ]
22%$2#&%2%&"#&%#&
%2%&"#&2%633#&&"68%#&%&#$&$687#&
%
2%633#&%%246#&%#&
:equationcorrector,he
%%
&%H&
H&&%
&%
==
=++=
++=
==
dx
dy
dx
dy
dx
dyh y y
dx
dy
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8- 32
[ ]
[ ]
&#%2%27#is %#&atof estimate,he
#iterationsthreeinconver'esal'orithmcorrectortheA'ain
%2%27#&22%&"#&&"68%#&%
&#$
&$687#&
%
22%&"#&%2%28#&%#&
%2%28#&22%$2#&&"68%#&%
&#$&$687#&
%
2%H&
H&2%
2%
%%H&
H&%%
=
=++=
++=
==
=++=
++=
x y
dx
dy
dx
dyh y y
dx
dy
dx
dy
dx
dyh y y
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8- 33
9ilne*(impson 9ethod
• 9ilnes equation is the predictor euqation#
• ,he (impsons rule is the corrector formula#
• 9ilnes equation !predictor):
-or the two initial samplin' points a one*step
method such as Eulers equation can be used#• (impsoss rule !corrector):
<%%=2
3
H%H&H
H2$&
−−
−+ +−+=iii
iidx
dy
dx
dy
dx
dyh y y
<3=2 H&H&
H&&
−+
−+ +++=ii ji
i jidx
dy
dx
dy
dx
dyh y y
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8- 34
Example 8*6: 9ilne*(impson 9ehtod
#3#& and 2#&atestimatewant to?e
&at&thatsuch :Groblem
==
===
x x y
x y y xdx
dy
& & &
&#& &#&$687 &#&"68%
&#% &#%2%27 %&"
Assume that we have the followin' values
obtained from the Euler*trape;oidal methodin Example 8*4#
x ydx
dy
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8- 36
[ ]
[ ]
2637%#&
&"68%#&)22%&"#&!3"%323#&2
&#$&$687#&
"%323#&2637&#&2#&
2637&#&
&"68%#&)22%&"#&!3"%3%3#&2
&#$&$687#&
"%3%3#&26363#&2#&
22
%2
%2
&2
=
+++=
==
=
+++=
==
y
dxdy
y
dx
dy
,he computations for x are complete#
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8- 37
,he 9ilne predictor equation for estimatin' y at x:
( )( ) ( )[ ]
"264%#&
&"68%#&%22%&"#&"%323#&%2
&#$3&
%%2
3
H&H%H2
H$$3
=
+−+=
+−+= dx
dy
dx
dy
dx
dyh
y y
624&$#&"264%#&3#&&3
==dx
dy
,he corrector formular:
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8- 38
( )[ ]
( )[ ]
complete#isit,hen
"267&#&22%&"#&"%323#&3624&6#&2
&#$%2%27#&
624&6#&"267&#&3#&
"267&#&
22%&"#&"%323#&3624$&#&2
&#$%2%27#&
3
2
%3
%3
H%H2$3
H%&3
=+++=
==
=
+++=
+++=
y
dxdy
dx
dy
dx
dy
dx
dyh y y
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8- 39
east*(quares 9ethod
• ,he procedure for derivin' the least*squares
function:
&# Assume the solution is an nth*order polnomial:
%# >se the boundar condition of the ordinar
differential equation to evaluate one of !bob&b%
Ibn)#2# Define the ob/ective function:
n
n x xb xbbb y ++++=
%
%&$J
dxe x∫ = %
dx
dy
dx
yd e −=
J where
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8- 40
3# -ind the minimum of - with respect to the unknowns
!b&b% b2Ibn) that is
"# ,he inte'rals in (tep 3 are called the normal
equationsK the solution of the normal equations ields
value of the unknowns !b&b% b2Ibn)#
∫ =
∂
∂=
∂
∂
xall ii
dx
b
ee
b
$%
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8- 41
Example 8*8: east*squares 9ethod
%F%
:solutionAnaltical
&#x$interval for theit(olve
$at&thatsuch :Groblem
xe y
x y xydx
dy
=
≤≤
===
&
&
$
&$
&$
J
&J
is modellinearthe,hus #& ields
)$!&J
condition boundarthe>sin'
J
bdx
yd
xb y
b
bb y
xbb y
=
+=
=
+==
+=• -irst assume a linear model is used:
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8- 42
$)"32
%
%!
$)&)<!&!=%%
&
)&!
:functionerror,he
$
"
&
32
&
%
&
$ $
%
&&
&
%
&
&&&
=++−−
=−+−=
−=
+−=−=
∫ ∫ x
x x
xb x xb x xb
dx x xb xbdxdb
dee
xdb
de
xb xb xybe
x y
x
x
2%&"
2%
&"
&
&J
,hus # b'etwe&withinte'ralabove
thesolve &$ran'etheininterestedarewe(ince
+=
==≤≤
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8- 45
( ) ( )[ ]( )
3
&
&%
"
&"
8
:limituppertheas & >sin'
$
34"%3
2
2
%
$&%&
&
%&
$
34
%
"
&
%3
%%
%
2
&&
$
%2
%
%
&
%
&
=+
=
=
+++−−+−
=−−−+−
−=∂
∂
∫
bb
x
x xb xb x xb xb
xb xb
dx x x x xb xb
xb
e
x
x
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8- 47
x ,rue y Lalue Bumerical y Lalue Error !)
$ &# &# *
$#% &#$%$% &#$$%% *
$#3 &#$822 &#$463 $#$
$#4 &#&76% &#&7"4 $#$
$#8 Ċ& 漠 $#$
&#$ ᄢ +&& $#$
,able: A quadratic model for the least*squares method
%F% xe y = %68664#$&3447#$&J x x y +−=
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8- 48
1alerkin 9ethod
• Example: 1alerkin 9ethod
,he same problem as Example 8*8#
>se the quadratic approximatin' equation#
i
i
i
x i
b
e!
!
niedx!
∂
∂=
==∫
method squaresleast-or the
factor# n'a wei'hti is where
###%& $
# and et %
%& x! x! ==
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8- 49
%
%&
%&
&
$
%2
%
%
&
&
$
2
%
%
&
8""%4#$%42&4#$&J
:result final ,he
3
&
2
&
&"
%
2
&
&"
6
&%
&
:equations normal followin' 'et the ?e
$<)%!)&!=
$<)%!)&!=
x x y
bb
bb
dx x x x xb xb
xdx x x xb xb
+−=
=+
=+
=−−+−
=−−+−
∫
∫
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8- 50
,able: Example for the 1alerkin method
x ,rue y value Bumerical y value Error!)
$ &# &# **
$#% &#$%$% $#78&4 $#$
$#3 &#$822 &#$2&4 $#$
$#4 &#&76% &#&"$$ $#$
$#8 Ċ& ࣈ $#$
&#$ ᄢ &#"7%& $#$
%F% xe y = %8""%4#$%42&4#$&J x x y +−=
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8- 51
Mi'her*5rder Differential Equations
• (econd order differential equation:
,ransform it into a sstem of first*order differential
equations#
dx
dy
dx
dy y y y
ydx
dy
y y x f dx
dy
===
=
=
&%&
%&
%&%
and where
)!
=
dx
dy y x f
dx
yd
%
%
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• .n 'eneral an sstem of n equations of the followin'
tpe can be solved usin' an of the previousl
discussed methods:
)###!
)###!
)###!
)###!
%&
%&22
%&%%
%&&&
nnn
n
n
n
y y y x f dx
dy
y y y x f dx
dy
y y y x f dx
dy
y y y x f dx
dy
=
=
=
=
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8- 53
Example: (econd*order Differential Equation
E" # #
E" $
d# % d
%
%
%
&$ :Groblem −==
& d#
d%
E" # #
E" $
d# d&
=
−==
&$
:into med transfor be can.t
%
h & % # f % %
h & % # f & &
& % # E"
iiiii
iiiii
)!
)!
:equations followin' thesolve tomethod s Euler+>se
$%2&3#$ and $$at24$$ Assume
&&
%&
+=+=
−====
+
+
bl d d iff i l i
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8- 54
,able: (econd*order Differential Equation
>sin' a (tep (i;e of $#& -t
X
(ft)
Y
(ft)
Exact Z Exact Y
(ft)
0 0 -0.0231481 0 -0.0231481 0
0.1 0.000275 -0.0231481 -0.0023148 -0.0231344 -0.0023144
0.2 0.0005444 -0.0231206 -0.0046296 -0.0230933 -0.004626
0.3 0.0008083 -0.0230662 -0.0069417 -0.0230256 -0.0069321
0.4 0.0010667 -0.0229854 -0.0092483 -0.0229319 -0.0092302
0.5 0.0013194 -0.0228787 -0.0115469 -0.0228125 -0.0115177
0.6 0.0015667 -0.0227468 -0.0138347 -0.0226681 -0.0137919
0.7 0.0018083 -0.0225901 -0.0161094 -0.0224994 -0.0160505
0.8 0.0020444 -0.0224093 -0.0183684 -0.0223067 -0.018291
0.9 0.002275 -0.0222048 -0.0206093 -0.0220906 -0.020511
d#
d&
d#
d% & =
, bl ( d d Diff i l E i
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,able: (econd*order Differential Equation
>sin' a (tep (i;e of $#& -t !continued)
d#
d&
d#
d% & =
X
(ft)
Y
(ft)
Exact Z Exact Y
(ft)
1 0.0025 -0.0219773 -0.0228298 -0.0218519 -0.0227083
2 0.0044444 -0.0185565 -0.0434305 -0.0183333 -0.04296663
3 0.0058333 -0.0134412 -0.0298019 -0.0131481 -0.0588194
4 0.0066667 -0.007187 -0.0704998 -0.0068519 -0.0688889
5 0.0069444 -0.0003495 -0.0746352 0.00000000 -0.071228
6 0.0066667 0.0065157 -0.0718747 0.0068519 -0.0688889
7 0.0058333 0.0128532 -0.06244066 0.0131481 -0.0588194
8 0.0044444 0.0181074 -0.0471107 0.0183333 -0.042963
9 0.0025 0.0217227 -0.0272183 0.0278519 -0.0227083
10 0.000000 0.0231435 -0.00466523 0.0231481 0.000000