C H A P T E R
5 Numerical Solution ofInitial-Value Problems
5.1 Introduction
Differential equations areused tomodel problems thatinvolve thechange ofsome variable
with respect toanother .These problems require thesolution toaninitial -value problem — 
thatis,thesolution toadifferential equation thatsatisfies agiven initial condition .
Inmany real-lifesituations ,thedifferential equation thatmodels theproblem istoo
complicated tosolve exactly ,andoneoftwoapproaches istaken toapproximate theso-
lution .Thefirstapproach istosimplify thedifferential equation toonethatcanbesolved
exactly ,andthen usethesolution ofthesimplified equation toapproximate thesolution
totheoriginal equation .Theother approach ,theoneweexamine inthischapter ,involves
finding methods fordirectly approximating thesolution oftheoriginal problem .This isthe
approach commonly taken because more accurate results andrealistic error information can
beobtained .
Themethods weconsider inthischapter donotproduce acontinuous approximation
tothesolution oftheinitial -value problem .Rather ,approximations arefound atcertain
specified ,andoften equally -spaced ,points .Some method ofinterpolation ,commonly cubic
Hermite ,isused ifintermediate values areneeded .
Thefirst partofthechapter concerns approximating thesolution y(t)toaproblem of
theform
d y— =f(t,y),fora<t<b,a t
subject toaninitial condition
y(a)=or.
These techniques form thecore ofthestudy because more general procedures usethese
asabase.Later inthechapter wedeal with theextension ofthese methods toasystem of
first-order differential equations intheform
dyi
dt=f\{ttyuy 2>.--,yn)>
dyi
dtf2(t*yu yi %-"*yn)%
dyn
dtfn(t,yuyi yn),
fora<t<b,subject totheinitial conditions
3Pi(a)=ai,y2(a)=ot2% yn(a)=ctn.
173
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Wealso examine therelationship ofasystem ofthistype tothegeneral nth-order initial -
value problem oftheform
yM=f(t,y,y\ y«-»)
fora<t<bysubject tothemultiple initial conditions
y(a)=of0,y\a)=y(n~x\a)=a„ -i.
Well -Posed Problems
Before describing themethods forapproximating thesolution toourbasic problem ,we
consider some situations thatensure thesolution willexist.Infact,because wewillnotbe
solving thegiven problem ,only anapproximation totheproblem ,weneed toknow when
problems thatareclose tothegiven problem have solutions thataccurately approximate the
solution tothegiven problem .This property ofaninitial -value problem iscalled well-posed ,
andthese aretheproblems forwhich numerical methods areappropriate .Thefollowing
result shows thattheclass ofwell-posed problems isquite broad .
Well-Posed Condition
Suppose that/and/y,itsfirstpartial derivative with respect toy,arecontinuous fort
in[a,b]andforally.Then theinitial -value problem
y'=/(f,y),fora <t<b,with y(a)=a,
hasaunique solution y(t)fora<t<b,andtheproblem iswell-posed .
Example 1Consider theinitial -value problem
y'=1+1sin(fy),for0<t<2,with y(0)=0.
Since thefunctions
/(/,y)=1+tsin(/y) and fy(t,y)=t2cos(/y)
areboth continuous for0<t<2andforally,aunique solution exists tothiswell-posed
initial-value problem .
Ifyouhave taken acourse indifferential equations ,youmight attempt todetermine
thesolution tothisproblem byusing oneofthetechniques youlearned inthatcourse .
5.2 Taylor Methods
Themethods inthissection use
Taylor polynomials andthe
knowledge ofthederivative ata
node toapproximate thevalue of
thefunction atanew node.Many ofthenumerical methods wesawinthefirstfourchapters have anunderlying deriva -
tion from Taylor ’sTheorem .The approximation ofthesolution toinitial -value problems
isnoexception .Inthiscase,thefunction weneed toexpand inaTaylor polynomial isthe
(unknown )solution totheproblem ,y(f).Initsmost elementary form ,thisleads toEuler’s
Method .Although Euler ’smethod isseldom used inpractice ,thesimplicity ofitsderiva -
tionillustrates thetechnique used formore advanced procedures ,without thecumbersome
algebra thataccompanies these constructions .
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Theuseofelementary difference
methods toapproximate the
solution todifferential equations
wasoneofthenumerous
mathematical topics thatwasfirst
presented tothemathematical
public bythemost prolific of
mathematicians .Leonhard Euler
(1707-1783 ).Theobjective ofEuler ’smethod istofind,foragiven positive integer N,anapproxi -
mation tothesolution ofaproblem oftheform
d y „ — =/(f,y),io ta <t<b,with y(a)=ad t
attheN+1equally -spaced mesh points {r0,t\,f2,  .f,v)(seeFigure 5.1).Thecommon
distance between thepoints ,h=(b— a)/N,iscalled thestep size ,and
/,=a4-ih,foreach j=0,1,...N.
Approximations atother values of/in[a,b]canthen befound using interpolation .
Figure 5.1
y
yM=y(b) y'=f(uy\^.y{a)=aS
yit^/
yit.)
y(to)=a
*0— a11 12    TH— b t
Suppose that y(t),thesolution totheproblem ,hastwocontinuous derivatives on[a,b\y
sothatforeach j=0,1,2,...,N— 1,Taylor ’sTheorem implies that
(*..yft+i)=yfo)+to+i-*>/<*>+--2— /(ft),
forsome number inft,f,+i).Letting h=(b-a)/N=f,+i-/j,wehave
h2
y(*+i>- +*/<M+y/(6).
and,since y(t)satisfies thedifferential equation y'(f)=/(r,y(r)),
yft+i)=y(t,)+A/ft.yfo))+yy"(fe).
Euler ’smethod constructs theapproximation w,toy(fj)foreach i=1,2,...,Nby
deleting theerror term inthisequation .This produces adifference equation thatapproxi -
mates thedifferential equation .Theterm local error refers totheerror atthegiven step if
itisassumed thatalltheprevious results areexact .The true,oraccumulated ,error ofthe
method iscalled global error .
Euler'sMethod
w0=a,
w,-+I=w,+h f(titw,),
where i=0,1,...,N—1,with local error\y"(%i)h2forsome £,in(titf,+j).
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Illustration InExample 1wewilluseEuler ’smethod toapproximate thesolution to
/=y-r2+l,0<r<2,y(0)=0.5,
att=2.Here wewillsimply illustrate thesteps inthetechnique when wehave h=0.5.
Forthisproblem /(/,y)=y— t2+1,so
w0=y(0)=0.5;
w,=w0+0.5(w0-(0.0)2+1)=0.5+0.5(1.5)=1.25;
w2=w\+0.5(wi-(0.5)2+1)=1.25+0.5(2.0)=2.25;
w3=w2+0.5(W2-(1.0)2+1)=2.25+0.5(2.25)=3.375 ;
and
y(2)»w4=w3+0.5(W3-(1.5)2-I-1)=3.375 +0.5(2.125 )=4.4375 .
Tointerpret Euler ’smethod geometrically ,note thatwhen w,isaclose approximation
toy(ti),theassumption thattheproblem iswell-posed implies that
Theprogram EULERM 51
implements Euler 's
method .fOi,Wf)y'(tj)=f(thyfc)).
The firststep ofEuler ’smethod appears inFigure 5.2(a),andascries ofsteps appears in
Figure 5.2(b).
Figure 5.2
y
wIa-y(a)-a
t0=a .. .t tNSlope y'(a)=f(a,a)
b t
(a)y
yip)--
vvv4-
w2+-y'=/0.y).
y(a)=a
to! t u v
i b lb l
Example 1Euler ’smethod wasused intheIllustration with h=0.5toapproximate thesolution tothe
initial -value problem
/=y_,2+l,0<l<2,y(0)=0.5.
Use theprogram EULERM 51with N=10todetermine approximations ,andcompare
these with theexact values given byy(t)=(t+l)2-0.5e'.
Solution With N=10wehave h=0.2,tx=0.2i,wo=0.5,and
wi+i=W i+h(w j— t f+l)=W i+0.2 [w i— 0.0412+l]=1.2w*— 0.008i2+0.2,
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fori=0,1,...,9.So
w,=1.2(0.5)— 0.008 (0)2+0.2=0.8;w2=1.2(0.8)-0.008 (1)2+0.2=1.152 ;
andsoon.Table 5.1shows thecomparison between theapproximate values atr,andthe
actual values .
u W i y,=y(ti) lyi-w/l
0.0 0.5000000 0.5000000 0.0000000
0.2 0.8000000 0.8292986 0.0292986
0.4 1.1520000 1.2140877 0.0620877
0.6 1.5504000 1.6489406 0.0985406
0.8 1.9884800 2.1272295 0.1387495
1.0 2.4581760 2.6408591 0.1826831
1.2 2.9498112 3.1799415 0.2301303
1.4 3.4517734 3.7324000 0.2806266
1.6 3.9501281 4.2834838 0.3333557
1.8 4.4281538 4.8151763 0.3870225
2.0 4.8657845 5.3054720 0.4396874
Error Bounds forEuler 'sMethod
Euler ’smethod isderived from aTaylor polynomial whose error term involves thesquare of
thestepsizeh,sothelocal error ateach stepisproportional to/r,soitis0(h2).However ,
thetotal error ,orglobal error,accumulates these local errors ,soitgenerally grows ata
much faster rate.
Euler'sMethod Error Bound
Lety(t)denote theunique solution totheinitial -value problem
y'=f(t,y), f o T a <t<b,withy (a)=a,
andwo,w i,...,WNbetheapproximations generated byEuler ’smethod forsome pos-
itive integer N.Suppose that/iscontinuous forallfin[a,b]andallyin(-oc,oc),
andconstants LandMexist with
<Land|y"(/)|<M.df
Then ,foreach i=0,1,2,...,TV,
ly(f.)-Wj|<— re«'.-°>-2L1].
Animportant point tonotice isthat,although thelocal error ofEuler ’smethod ,thatis,
theerror atanindividual step,is0(h2),theglobal error,which istheerror over theentire
interval ,isonly O(h).Thereduction ofonepower ofhfrom local toglobal error istypical
ofinitial-value techniques .Even though wehave areduction inorder from local toglobal
errors ,theformula shows thattheerror tends tozero with h.
Example 2Thesolution totheinitial -value problem
y'=y-t2+1,0<r<2,y(0)=0.5,
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wasapproximated inExample 1using Euler ’smethod with h=0.2.Find bounds forthe
approximation errors andcompare these totheactual errors .
Solution Because fit,y)=y-12+1,wehave 9fit,y)/dy=1forally,soL— 1.For
thisproblem ,theexact solution isyit)=it+l)2-0.5e\so/'(0=2— 0.5e'and
\y"(t)\<0.5e2-2,forall/6[0,2].
Using theinequality intheerror bound forEuler ’smethod with h=0.2,L=1,and
Af=0.5e2-2gives
|yi-wf\<0.1(0.5e2-2)(«?"-1).
Hence
|>(0.2)-wH<0.1(0.5e2-2)(e02-1)=0.03752 ;
|y(0.4)-w21<0.1(0.5e2-2)(e04-1)=0.08334 ;
andsoon.Table 5.2lists theactual error found inExample 1,together with thiserror
bound.Note thateven though thetrue bound forthesecond derivative ofthesolution was
used,theerror bound isconsiderably larger than theactual error,especially forincreasing
values oft. m
Table 5.2
ti 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Actual Error 0.02930 0.06209 0.09854 0.13875 0.18268 0.23013 0.28063 0.33336 0.38702 0.43969
Error Bound 0.03752 0.08334 0.13931 0.20767 0.29117 0.39315 0.51771 0.66985 0.85568 1.08264
Higher Order Taylor Methods
Euler ’smethod wasderived using Taylor ’sTheorem with n=1,sothefirstattempt tofind
methods forimproving theaccuracy ofdifference methods istoextend thistechnique of
derivation tolarger values ofn.Suppose thesolution yit)totheinitial-value problem
y'=f it,y),ior a <t<b,withy (a)=of,
hasn+1continuous derivatives .Ifweexpand thesolution y(f)interms ofitsnthTaylor
polynomial about f,,weobtain
h2hn hn+*yfo+i)=yOi)+h y\u)+— y” (ii)+   +— y^ft)+ ....y("+l)(fe)2 n\ in+1)!
forsome number £,in(/,, Successive differentiation ofthesolution y(f)gives
y'(t)=fit,y(0).y"(t)=fit,yit)),and,generally ,y{k)it)=fk~l)(t,yit) ).
Substituting these results intotheTaylor expansion gives
h2
yfe+i)=yM+hfitityiti ))+y/'fa,yM )+   
hn /,*+1
+ yfe))+— — r/Wfc.y®)).n! in+1)!
Thedifference -equation method corresponding tothisequation isobtained bydeleting
theremainder term involving
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Taylor Method ofOrder n
W0=Of,
Wr+l=w,+hT w (fi,W i)
foreachi=0,1,...,N— 1,where
W i)=/«,.wi)+JfXh,w,)+   + W i).2 nl
The local error is^^-y(n+1)(f,)/i',+1forsome £,in(/,,f,+i).
Theformula forT(n)iseasily expressed butdifficult tousebecause itrequires the
derivatives of/with respect tot.Since/isdescribed asamultivariable function ofboth
tandy,thechain ruleimplies thatthetotal derivative of/with respect tot,which we
denoted /'(r,y(f)),isobtained by
r o.y(t))=g(r,*0) £+ =ga,y(0)+
or,since y'(r)=/(f,y(0),by
/'(f.y(0)=^(',y(0)+/(f,y(0)%(fy(0).3/ 3y
Higher derivatives canbeobtained inasimilar manner ,butthey might become increasingly
complicated .Forexample ,/"(/,y(f))involves thepartial derivatives ofalltheterms on
theright side ofthisequation with respect toboth tand y.
Example 3Apply Taylor ’smethod oforders (a)2and(b)4with N=10totheinitial -value problem
y'=y-t2+\,0<r<2,y(0)=0.5.
Solution (a)Forthemethod oforder 2weneed thefirstderivative of/(f,y(f))=y(t)-
t2+1with respect tothevariable t.Because y'=y— t2+1wehave
f(t,y(t) )=U y-t2+l)=y'-2t=y-t2+l-2t,at
so
Ti2)(tit =/(/,,w^+
^/'(*,,=Wi-tf+1+
^(w,-t2+1-2f|)
=(l+ (w,~t?+1)-h t i.
Because A=10wehave h=0.2,and /,=0.2/foreach i=1,2,....10.Thus the
second -order method becomes
w0=0.5,
W/+I=W i+h
^1+^(w,-t f+1)-h t,
=w,+0.2|^1+ ("i-0.04 /2+1)-0.04i
=1.22w,-0.0088 /2-0.008 /+0.22.
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Table 5.3
tiTaylor
Order 2
w,Error
lyft)-wi\
0.0 0.500000 0
0.2 0.830000 0.000701
0.4 1.215800 0.001712
0.6 1.652076 0.003135
0.8 2.132333 0.005103
1.0 2.648646 0.007787
1.2 3.191348 0.011407
1.4 3.748645 0.016245
1.6 4.306146 0.022663
1.8 4.846299 0.031122
2.0 5.347684 0.042212Thefirsttwosteps give theapproximations
y(0.2)asvvi=1.22(0.5)-0.0088 (0)2-0.008 (0)+0.22=0.83;
y(0.4)a;w2=1.22(0.83 )-0.0088 (0.2)2-0.008 (0.2)+0.22=1.2158 .
Alltheapproximations andtheir errors areshown inTable 5.3.
(b)ForTaylor ’smethod oforder 4weneed thefirstthree derivatives off(t,y(t))with
respect tot.Again using y'=y— t2-f1wehave
/'(*,y(0)=y-12+1-2t,
f"(t,y(r))=4-(y-t2+1-20=/-2»-2=y-12+1-2r-2at
=y-12-2/-1,
and
/"'(*.y(0)=^(y— *2—2r—1)=/— 2r— 2=y— t2— 2r—1,
SO
T(4)(tt,w4)=/ft,w,)+
^/'ft,Wf)+
^r/"ft,w4)+^/'"ft,w,-)
=w,-— tf+1+-(w,-tf+1-2*i)+— (w,— tf— 2ti— l)
h h2/i3\f 2\ft h^2\/f..h h2
Hence Taylor ’smethod oforder 4is24*
w0=0.5,—^+y(*,-<?)-(i+1+1+5^hT
2 4
fori=0,1 tf-1.
Because N=10andh=0.2themethod becomes
Table 5.4
tiTaylor
Order 4
w,Error|yfo)-w,|
0.0 0.500000 0
0.2 0.829300 0.000001
0.4 1.214091 0.000003
0.6 1.648947 0.000006
0.8 2.127240 0.000010
1.0 2.640874 0.000015
1.2 3.179964 0.000023
1.4 3.732432 0.000032
1.6 4.283529 0.000045
1.8 4.815238 0.000062
2.0 5.305555 0.000083wj+1=Wj+0 .2[(l+^(Wi-0.04i2)
004\-(1+=+-jj-J(0.04i)+1+002
302
20.04 0.008
6 24
=1.2214*,--0.008856 /2-0.00856 /+0.2186 ,
foreach i=0,1,...,9.Thefirsttwosteps give theapproximations
y(0.2)asH»,=1.2214 (0.5)-0.008856 (0)2-0.00856 (0)+0.2186 =0.8293 ;
y(0.4)asW2=1.2214 (0.8293 )-0.008856 (0.2)2-0.00856 (0.2)+0.2186 =1.214091 .
Alltheapproximations andtheir errors arcshown inTable 5.4.
Compare these results with those ofTaylor ’smethod oforder 2inTable 5.3andyou
willseethattheorder 4results arevastly superior .
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Approximating Intermediate Results
Hermite interpolation requires
both thevalue ofthefunction and
itsderivative ateach node.This
makes itanatural interpolation
method forapproximating
differential equations because
these dataareallavailable .Theresults from Table 5.4indicate theTaylor ’smethod oforder 4results arequite accurate
atthenodes 0.2,0.4,etc.Butsuppose weneed todetermine anapproximation toan
intermediate point inthetable,forexample ,at/=1.25.Ifweuselinear interpolation on
theTaylor method oforder four approximations att=1.2andt=1.4,wehave
/1.25)«= )3'179964 +V M-1^j3'732432 =3.318081 .
The true value isy(1.25)=3.317329 ,sothisapproximation hasanerror of0.000752 ,
which isnearly 30times theaverage oftheapproximation errors at1.2and1.4.
Wecansignificantly improve theapproximation byusing cubic Hermite interpolation .
Todetermine thisapproximation fory(1.25)requires approximations toy'(1.2)andy'(1.4)
aswellasapproximations toy(1.2)andy(1.4).However ,theapproximations fory(1.2)and
y(l.4)arcinthetable ,andthederivative approximations areavailable from thedifferential
equation because y'(t)=f(t,y(t)).Inourexample y'(t)=y(f)— t2+1,so
/(1.2)=/1.2)-(1.2)2+1«=3.179964 -1.44+1=2.739964
and
/(1.4)=/1.4)-(1.4)2+1«3.732433 -1.96+1=2.772432 .
The divided -difference procedure inSection 3.4gives theinformation inTable 5.5.
Theunderlined entries come from thedata,andtheother entries usethedivided -difference
formulas .
Table 5.512 3.179964
2.739964
1.2 3.179964 0.111880
2.762340 -0.307100
1.4 3.732432 0.050460
2.772432
1.4 3.732432
Thecubic Hermite polynomial is
/0«=3.179964 +(t-1.2)2.739964 +(t-1.2)20.111880
+(f-1.2)2(r-1.4)(— 0.307100 ),
so
y(1.25)«3.179964 -I-0.136998 +0.000280 +0.000115 =3.317357 ,
aresult thatisaccurate towithin 0.000028 .This isabout theaverage oftheerrors at1.2
andat1.4,andonly 4%oftheerror obtained using linear interpolation .This improvement
inaccuracy certainly justifies theadded computation required fortheHermite method .
Error estimates fortheTaylor methods aresimilar tothose forEuler ’smethod .If
sufficient differentiability conditions aremet,annth-order Taylor method willhave local
error0(hn~x)andglobal error0(hn).
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E X E R C I S E S E T 5 2
1. UseEuler ’smethod toapproximate thesolutions foreach ofthefollowing initial -value problems .
a.y'— teyi-2y,for0<t<1,with y(0)=0andh— 0.5
b.y=14-(/— y)2,for2<t<3,with y(2)=1andh=0.5
c.ys1+y,for1</<2,with y(l)=2andh=0.25
d.y'=cos21+sin3/,for0<t<1,with y(0)=1andh=0.25
2. The actual solutions totheinitial -value problems inExercise 1aregiven here.Compare theactual
error ateach step totheerror bound .
a.y(t)=
^/e3'-^£314-^eb .y(/)=t+(1-t)~l
1. 1 4c.y(/)=/In/4-2/ d.y(t)=-sin2/— -cos3/-I--
w J J
3. UseEuler’smethod toapproximate thesolutions foreach ofthefollowing initial -value problems .
a.y=j— ,for1<t<2,with y(l)=1andh=0.1
b.y'=14-~4-(
^),for1</<3,with y(l)=0andh=0.2
c.y'— -(y+l)(y4-3),for0<t<2,with y(0)=-2andh=0.2
d.y=— 5y4-5/24-2/,for0</<1,with y(0)=1/3andh=0.1
4. The actual solutions totheinitial -value problems inExercise 3aregiven here.Compute theactual
error intheapproximations ofExercise 3.
a.y(t)=/(14-In/)"1
b.y(/)=/tan(ln/)
c. y(t)=— 34-2(14-e-2/)-1
d.y(/)=t24-
5. Repeat Exercise 1using Taylor ’smethod oforder 2.
6. Repeat Exercise 3using Taylor ’smethod oforder 2.
7. Repeat Exercise 3using Taylor ’smethod oforder 4.
8. Given theinitial-value problem
y'=j y+l2e',1<I<2.y(l)=0
with theexact solution y(f)=t2(e'— e):
a.UseEuler ’smethod with h=0.1toapproximate thesolution andcompare itwith theactual
values ofy.
b.Usetheanswers generated in(a)andlinear interpolation toapproximate thefollowing values of
yandcompare them totheactual values .
i. y(1.04) ii. y(1.55) iii. y(1.97)
c.UseTaylor ’smethod oforder 2with h=0.1toapproximate thesolution andcompare itwith
theactual values ofy.
d.Usetheanswers generated in(c)andlinear interpolation toapproximate yatthefollowing values
andcompare them totheactual values ofy.
i. y(1.04) ii. y(1.55) iii. y(1.97)
e.UseTaylor ’smethod oforder 4with h=0.1toapproximate thesolution andcompare itwith
theactual values ofy.
Copyright 2012 Cc«£»fcLearn in*.AIR.(huReversed Mayr*xbecopied ,canned ,o*duplicated.»whole oempan .Doctoelectronic rifhu.vomc third pony content may bevupprcv «dftem theeBook and/orcCh<xcn»l.Editorial roiew h*>
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f.Usetheanswers generated in(e)andpiecewise cubic Hermite interpolation toapproximate yat
thefollowing values andcompare them totheactual values ofy.
i. y(1.04) ii.>(1.55) iii.>(1.97)
9. Given theinitial -value problem
y'=}i-*-y2'i<»<2.:y(U=-i
with theexact solution >(/)=-1/f.
a.UseEuler ’smethod with h=0.05 toapproximate thesolution andcompare itwith theactual
values of>.
b.Usetheanswers generated in(a)andlinear interpolation toapproximate thefollowing values of
>andcompare them totheactual values .
i.>(1.052 ) ii.>(1.555 ) iii.>(1.978 )
c.UseTaylor ’smethod oforder 2with h=0.05 toapproximate thesolution andcompare itwith
theactual values of>.
d.Usetheanswers generated in(c)andlinear interpolation toapproximate thefollowing values of
>andcompare them totheactual values.
i.>(1.052 ) ii.>(1.555 ) iii.>(1.978 )
e.UseTaylor ’smethod oforder 4with h=0.05 toapproximate thesolution andcompare itwith
theactual values of>.
f.Usetheanswers generated in(e)andpiecewise cubic Hermite interpolation toapproximate the
following values of>andcompare them totheactual values .
i.>(1.052 ) ii.>(1.555 ) iii.>(1.978 )
10. Inanelectrical circuit with impressed voltage £,having resistance R,inductance L,andcapacitance
Cinparallel ,thecurrent isatisfies thedifferential equation
di d2£ 1d£ 1
dt~C~d^+R~
dt+L£'
Suppose i(0)=0,C=0.3farads ,R=1.4ohms ,L=1 . 7henries ,andthevoltage isgiven by
£(t)=eoab"sin(2f-jr).
UseEuler ’smethod tofindthecurrent iforthevalues t=0.1j,j=0,1,...,100.
11. Aprojectile ofmass m=0.11 kgshot vertically upward with initial velocity u(0)=8m/sisslowed
duetotheforce ofgravity FK=mgandduetoairresistance Fr=— w h e r e g=— 9.8m/s2
andk=0.002 kg/m.Thedifferential equation forthevelocity t;isgiven by
mu'=mg-&v|v|.
a.Find thevelocity after 0.1,0.2,...,1.0s.
b.Tothenearest tenth ofasecond ,determine when theprojectile reaches itsmaximum height and
begins falling.
5.3 Runge -Kutta Methods
Inthelastsection wesaw how Taylor methods ofarbitrary high order canbegenerated .
However ,theapplication ofthese high-order methods toaspecific problem iscomplicated
bytheneed todetermine andevaluate high-order derivatives with respect totontheright
side ofthedifferential equation .The widespread useofcomputer algebra systems has
simplified thisprocess ,butitstillremains cumbersome .
Copyright 2012 Cengagc Learn in*.AIRights Reserved Mayr*xbecopied.scanned .o*implicated ,inwhole orinpar.Doctocjectronie rifhu.*wethird pur.ycontent may besupprc-icdftem theeBook and/orcChaptcnM .Editorial roiew h*>
deemed CutanyMpprcucd content does notmaterialy alTcct theoverall Icamir*experience .Ceagage l.cammu reserves theright*>remove additional conceal atanytime ifvutoeqjeni nghtv restrictions require It