214 C H A P T E R 5 Numerical Solution ofInitial -Value Problems
2. Theinitial -value problem
y=—y+1— j,for1<t<2,with y(l)=1
hastheexact solution y(f)=14-(el~‘— l)/-1.
a.UsetheRunge -Kutta-Fehlberg method with tolerance TOL=10~3tofind wiandw^.Compare
theapproximate solutions totheactual values .
b.Use theAdams Variable -Step-Size Predictor -Corrector method with tolerance TOL=0.002
andstarting values from theRunge -Kutta method oforder 4tofind*v4andw5.Compare the
approximate solutions totheactual values.
3. UsetheRunge -Kutta -Fehlberg method with tolerance TOL=104toapproximate thesolution tothe
following initial-value problems .
a.y'= for1<t<1.2,with y(l)=1,hmax =0.05,andhmin=0.02.
b.y=sinf+e~\for0<t<1,with y(0)=0,hmax =0.25 ,andhmin=0.02.
c.y'— (y2+y)t~l,for1<t<3,with y(l)=— 2,hmax— 0.5,andhmin— 0.02.
d.y=— ty+4ty~\for0<t<1,with y(0)=1,hmax=0.2,andhmin=0.01.
4. UsetheRunge -Kutta -Fehlberg method with tolerance TOL=106,hmax =0.5,andhmin =0.05
toapproximate thesolutions tothefollowing initial -value problems .Compare theresults totheactual
values .
a.y=j- for1<t<4,with y(l)=1;actual solution y(t)=f/(l4-Inf).
b.y=1+j+ ,for1<t<3,with y(l)=0;actual solution y{t)=ttan(lnf).
c.y=— l)(>-l-3),for0<f<3,with>(0)=-2;actual solution .y(/)=— 3+2(1+e-2,)_
l.
d.y=(f+2/3)y3-f)\for0<t<2,with y(0)= actual solution y(t)=(3+2f2+6«/*)“ ,/2.
5. UsetheAdams Variable -Step-Size Predictor -Corrector method with TOL=104toapproximate the
solutions totheinitial-value problems inExercise 3.
6. UsetheAdams Variable -Step-Size Predictor -Corrector method with tolerance TOL=10~6,hmax =
0.5,andhmin=0.02 toapproximate thesolutions totheinitial-value problems inExercise 4.
7. Anelectrical circuit consists ofacapacitor ofconstant capacitance C=1.1farads inseries with a
resistor ofconstant resistance Ro=2.1ohms .Avoltage E(t)=110sin/isapplied attime t=0.
When theresistor heats up,theresistance becomes afunction ofthecurrent i,
R(t)=R0-fki,where k=0.9,
andthedifferential equation foribecomes
/1+2*.\d«+_1__.=\dS
y R{))dt RQC R0Cdt
Find thecurrent iafter 2s,assuming i(0)=0.
5.7Methods forSystems ofEquations
The most common application ofnumerical methods forapproximating thesolution of
initial -value problems concerns notasingle problem ,butalinked system ofdifferential
equations .Why,then,have wespent themajority ofthischapter considering thesolution
ofasingle equation ?Theanswer issimple :toapproximate thesolution ofasystem of
initial -value problems ,wesuccessively apply thetechniques thatweused tosolve problems
involving asingle equation .Asissooften thecase inmathematics ,thekeytothemethods
Cop>?i£tu2012 Cc«£»fcLearn in*.AIR.(huRocncd Mayr*xbecopied ,canned ,o*daplicaied.»whole o»mpan.Doctoelectronic rifhu.*wcthird pony contcac may besuppreved Trent theeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed CutanyMpprcucd content dec>notimtctlaly alTca theoverall leamir*experience .C'cttitape Learnmp re\me»thertpht 10renxyve additional conceal atanytime i iwtoeqjcni npht »rotrictionc require It5.7 Methods forSystems ofEquations 215
forsystems canbefound byexamining theeasier problem andthen logically modifying it
totreat themore complicated situation .
Anmth-order system offirst-order initial -value problems hastheform
dui
dt
du2
Itf\(*,U\,U2,
^**1»**2»   »**m)»
fora<t<b,with theinitial conditions
u\(a)=au u2(a)=a2,. ...um(a)=otm.
Theobject istofindmfunctions u\tu2,.. .,umthatsatisfy thesystem ofdifferential
equations together with alltheinitial conditions .
Methods tosolve systems offirst -order differential equations aregeneralizations ofthe
methods forasingle first-order equation presented earlier inthischapter .Forexample ,the
classical Runge -Kutta method oforder 4given by
w0=cr,
*i=hfOh wi)»
k2=hf
^ti-I- w,+
*3=hf\
^tj+ Wi+ ,
*4=hffa+uWi+*3),
and
w,+1=w,+~[k\+2k2+2ki+*4],o
foreach i=0,1,.. .,N— 1,isused tosolve thefirst-order initial -value problem
y'=f(t,y),fora<t<b,with y(a)=or.
Itisgeneralized forsystems asfollows .
Letaninteger N>0bechosen and seth=(b— a)/N.Partition theinterval [a,b]
into Nsubintervals with themesh points
t j=a+jh foreach j=0,1,...,N.
Use thenotation w yforeach j=0,1,. . .,Na n d i=1,2,. . .,mtodenote an
approximation to!*,(*,);thatis,w yapproximates theithsolution U j(t)ofthesystem atthe
y'thmesh point t j.Fortheinitial conditions ,set
wi,0=oi. W2,O=C*2,....wm>0=ofm.
Figure 5.4gives anillustration ofthisnotation .
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Figure 5.4
1
1
1
1
—>\
nmy
-
 "23--"«\ (a)=U'(0"’ 'i, y,
:s ^1-
u2(a)=a2i
 
a~*o*\hh1 a~*o hh* a— tot\*2h*
Suppose thatthevalues w2jt  ,wm.jhave been computed .Weobtain w\j+\,
W2j+u-  »wm,j+ibyfirstcalculatings foreach i=1,2,...,m,
k\j=hfi(tjtW\tj%W2J Wmj ),
andthen finding ,foreach i,
, h 1, 1, 1,\*2.»=hfiitj +-,Wij+-k\titW 2,j+2*1.2.   »Wm,j+2*1.1«) 
Wenext determine alltheterms
*3.i=hfi
^tj+ W\j+“ *2.1,w2,j+2^2.2.   .WmJ+-jk2,m
and,finally ,calculate alltheterms
*4,i=hfi(tj+h,W\j+*3.1."2.;+*3.2.   ."m./+*3.m)-
Combining these values gives
Theprogram RK04SY57
implements the
Rungc-Kutta method of
order 4forsystems ."i,y+1=wiJ+
^[*l,i+2*2.I+ <+*4.i]
foreach i=1,2,...m.
Note thatallthevalues*1.1,*1.2*-- .*i.mmust becomputed before anyoftheterms
oftheform*2,/canbedetermined .Ingeneral ,each*/.1**/.2»   .*/.mmust becomputed
before anyoftheexpressions*/+itl.
Example 1Kirchhoff ’sLaw states thatthesum ofallinstantaneous voltage changes around aclosed
electrical circuit iszero.This implies thatthecurrent ,/(/),inaclosed circuit containing a
resistance ofRohms ,acapacitance ofCfarads ,aninductance ofLhenrys ,andavoltage
source ofE(t)volts must satisfy theequation
LI\t)+RI(t)+^jI(t)dt=E(t).
Thecurrents I\(t)andl2{t)intheleftandright loops ,respectively ,ofthecircuit shown in
Figure 5.5arcthesolutions tothesystem ofequations
2/,(0+6[/1(0-/2(0]+2/i'(0=12,
LJi2(t)dt+4/2(0+6[/2(0-/1(»)]=0.
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deemed Cutany Mjppic-cdcontent does notmaxrUXy alTcct theiAera .1learning experience .('engage [.camonreserve,theright 10remove additional eonteatatanytime iisubvcqjem right,restriction,require It.Figure 5.55.7 Methods forSystems ofEquations 217
211 0.5F
K i— ^11.(0
6(I 412 12V— 
2H
Suppose that theswitch inthecircuit isclosed attime t=0.This implies that/,(0)
and/2(0)=0.Solve for/((/)inthefirstequation ,differentiate thesecond equation ,and
substitute forI[(t)toget
/,'=/,(f,/,,/2)=-4/,+3/2+6,with/,(0)=0,
/2=/2O.h J i )=0.6/,'-0.2/2=-2.4/,+I.6/2+3.6,with/2(0)=0.
Theexact solution tothissystem is
/,(t)=-3.375 e-2'+1.87V0-*+1.5 and /2(r)=-2.25e“ 2'+2.25e-°-4'.
Wewillapply theRunge -Kutta method oforder 4tothissystem with h=0.1.Since
Who=/.(0)=0andw2.o=/2(0)=0,
*1.1=*/ifo.W1.0,*2.0)=o.l/,(0,0,0)=0.1[-4(0)+3(0)+6]=0.6,
*i.2=hf2(to,wi.o,w2.o)=0.1/2(0.0,0)=0.1[-2.4(0)+1.6(0)+3.6]=0.36,
*2.1=*/i(to+2^’wi.o+2*1.1*w2.o+2*i.2
^=0*1/i(0.05,0.3,0.18 )
*2,2=hf20.1[— 4(0.3)+3(0.18 )+6]=0.534 ,
1(to+
^*»W|,o+-*1.1 »^2,0+2*1.2
^0.1/2(0.05,0.3,0.18 )
0.1[— 2.4(0.3)+1.6(0.18)+3.6]=0.3168 .
Generating theremaining entries inasimilar manner produces
*3f,=(0.1)/,(0.05 ,0.267 ,0.1584 )=0.54072 ,
*3.2=(0.1)/2(0.05 ,0.267 ,0.1584 )=0.321264 ,
*4.1=(0.1)/,(0.1,0.54072 ,0.321264 )=0.4800912 ,
*4.2=(0.1)/2(0.1,0.54072 ,0.321264 )=0.28162944 .
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Asaconsequence ,
/1(0.1)«wu=wi ,o+ +2*2,I+2*3.1+*4.i]
=0+-[0.6+2(0.534 )+2(0.54072 )+0.4800912 ]=0.53825526
and
/2(0.1)%w2,i=w2,o+g[*i,2+2*2.2+2*3,2+^4.2]=0.3196263 .
Theremaining entries inTable 5.15 aregenerated inasimilar manner .
Table 5.15t j W UW2J \h(t j)-w2j\
0.0 0 0 0 0
0.1 0.5382550 0.3196263 0.8285 xlO'50.5803 x10“ 5
0.2 0.9684983 0.5687817 0.1514 x1040.9596 x10~5
0.3 1.310717 0.7607328 0.1907 x1040.1216 x104
0.4 1.581263 0.9063208 0.2098 x10~40.1311 x104
0.5 1.793505 1.014402 0.2193 xlO"40.1240 x10~4
Any ofthemethods implemented inMATLAB canbeused forsystems ofdifferential
equations .Forexample ,touseode45tosolve oursystem given inExample 1wefirstdefine
theright -hand sides using anM-filecalled F.mthatcontains thestatements
function dy=F(t,y)
dy=zeros (2,1);
dy(1)=-4*y(1)+3*y(2)+6;
dy(2)=-2 . 4*y(l)+l.6*y(2)+3 . 6;
Then make theright -hand sideofthesystem ofdifferential equations known toMATLAB
with
FF=OF
Wenow define thetvalues atwhich wewant toapproximate thesolutions
tspan =[00.1 0.2 0.30.4 0 . 5]
The following command computes thesolution tothesystem atthegiven values oft.The
initial conditions I\(0)=0and/2(0)=0aregiven as[00].
[T,YY]=ode45(FF,tspan ,[00])
The MATLAB response places thetvalues inthearray Tand theapproximate solution
values inYY,with theapproximations forI\(t)inthefirstcolumn and/2(f)inthesecond .
0
0.100000000000000
0.200000000000000
0.300000000000000
0.400000000000000
0.5000000000000000
0.538263922676270
0.968513005638230
1.310736555252393
1.581284356153020
1.7935270443890290
0.319632054268176
0.568791683477228
0.760744806883952
0.906333359733513
1.014415449337470
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Higher -Order Differential Equations
Many important physical problems — forexample ,electrical circuits andvibrating systems — 
involve initial -value problems whose equations have order higher than 1.New techniques
arenotrequired forsolving these problems .Byrelabeling thevariables wecanreduce
ahigher -order differential equation intoasystem offirst-order differential equations and
then apply oneofthemethods wehave already discussed .
Ageneral mth-order initial -value problem hastheform
/">(»)=/(»,*/y—'o,
fora<t<b,with initial conditions
y(a)=ctuy'(a)=a2 y(m~[)(a)=am.
Toconvert thisintoasystem offirst-order differential equations ,define
mW=yd).u2(t)=y\t) um(t)=/—»«.
Using thisnotation ,weobtain thefirst-order system
duidy— =— =“ 2,dt dt
dui
dtdy
dt=w3,
dum— \
dtdy(m— 2)
dt=Um»
and
dum
dtdy(ml)
=v(m)
dty y{ml))=f(t,u u u 2 u m).
with initial conditions
u,(a)=y(a)=or,,u2(a)=y’(a)=a2 um(a)=y(ml)(a)=am.
Example 2Transform thesecond -order initial -value problem
y” — 2y'+2y=e*sinf, forO <t<1,with y(0)=— 0.4,>'(0)=— 0.6
intoasystem offirstorder initial -value problems ,andusetheRunge -Kutta method oforder
4with h=0.1toapproximate thesolution .
Solution Let wi(f)=y(f)and 112(f)=/(/).This transforms thesecond -order equation
intothesystem
*<5(0=«2(f)»
u'2(t)=e2tsinf-2u\(t)+2u2(t),
with initial conditions wj(0)=-0.4,u2(0)=— 0.6.
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The initial conditions give wio=— 0.4and w2io=— 0.6.The Runge -Kutta method
oforder 4forsystems described onpage 216with j=0give
*i.i=Vi(lb,wit0,w2,o)=hw 2,o=-0.06 ,
*1.2=V2fo>.wi.o,w2.o)=hlf*sin/o-2w1>0+2w2>0]=-0.04 ,
*2.1=Vl
*2*[*2.0+-*1.2( h 1 1.\Uo+2»wi.o+2*1.1 »w2.o+2*1.2J=
,2=*/2
^*0+ Wi,o+-*1.1»2,0+2*1>2
^ =hj^*005*sin(r0+0.05 )-2
^wi.0+
^*1,1
^+2
^w2,0+
^*1.2
^j-0.062 ,
=-0.03247644757 ,
*3,1=*w2,o+
^*2,2-0.06162832238 ,
e2(r0-o.o5)sin^+Q05)-2
^wi.o+
^*2.1
^+2
^w2,o+2*2-2)] *3,2=h
=-0.03152409237 ,
*4.1=h[w2.o+*3.2]=-0.06315240924 ,
and
*4.2=*[e2(tQ+0A)sin(lb+0.1)-2(w,
t0+*3.1)+2(w2.0+*3.2)]=-0.02178637298 .
So
Wj.i=W|,o+— (*1,1+2AT2.1+2*3.1“ H*4,1)=— 0.4617333423 and6
W2.1=w2.o+|(*,.2+2*2.2+2*3.2+*«)=-0.6316312421 .
The value wj.iapproximates wj(0.1)=y(0.1)=0.2e2(OI)(sin0.1— 2cos0.1),and
W2.1approximates «2(0.1)=>'(0.1)=0.2^2(0,)(4sin0.1— 3cos0.1).
The setofvalues w\jand w>2jtfor7=0,1,. ..»10,arepresented inTable 5.16 and
arecompared totheactual values of«i(/)=0.2^2,(sint — 2cos/)and«2(r)=«!(/)=
0.2e21(4sin/— 3cos/).
Table 5.16
tj yitj)=u1(tj)WUy%)=u2(tj)” 2J
0.0 <B1P!7!Tr?!S-0.40000000 -0.60000000 BTSTiTiTW 0 0
0.1 -0.46173297 -0.46173334 -0.63163105 -0.63163124 3.72x10-71.92 x10-7
0.2 -0.52555905 -0.52555988 -0.64014866 -0.64014895 8.36 x10-72.84 x10-7
0.3 -0.58860005 -0.58860144 -0.61366361 -0.61366381 1.39x10-61.99 x107
0.4 -0.64661028 -0.64661231 -0.53658220 -0.53658203 2.02x10-61.68 x10-7
0.5 -0.69356395 -0.69356666 -0.38873906 -0.38873810 2.71x10“ 9.58 x10~7
0.6 -0.72114849 -0.72115190 -0.14438322 -0.14438087 3.41x10"62.35 x106
0.7 -0.71814890 -0.71815295 0.22899243 0.22899702 4.06 x10"64.59 x10"6
0.8 -0.66970677 -0.66971133 0.77198383 0.77199180 4.55x10"67.97 x10"6
0.9 -0.55643814 -0.55644290 1.5347686 1.5347815 4.77x10-61.29 x105
1.0 -0.35339436 -0.35339886 2.5787466 2.5787663 4.50x10~61.97 xIQ5
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Other one-stepapproximation methods canbeextended tosystems .IftheRunge -Kutta -
Fehlberg method isextended ,then each component ofthenumerical solution w\j,w2j>..
wmjmust beexamined foraccuracy .Ifanyofthecomponents failtobesufficiently accurate ,
theentire numerical solution must berecomputed .
Themulti step methods andpredictor -corrector techniques canalso beextended easily
tosystems .Again ,iferror control isused,each component must beaccurate .Theextension
oftheextrapolation technique tosystems canalso bedone ,butthenotation becomes quite
involved .
E X E R C I S E S E T 5 . 7
1. Use theRunge -Kutta method oforder 4forsystems toapproximate thesolutions ofthefollowing
systems offirst-order differential equations andcompare theresults totheactual solutions .
a.u\=3i*|4-2M2— (2/2+lie2',f o r O </<lwith «,(0)=l;
«2=4u\4-u24-(/2-1-2/— 4)e2',for0</<1with «2(0)=1;
h=0.2;actual solutions «!(/)=\e5‘- +e21and «2(/)=\e51+\e~*4-t2e21.
b. — 4«j— 2M2+cosr -f4sinr,f o r0</<2with «!(())=0;
«2=3«i4-«2-3sin/,for0<t<2with «2(0)=— 1;
h=0.1;actual solutions uj(f)=2e~l— 2e-2/4-sin/and u2(/)=— 3e-'4-2e-2/.
c.u\=«2,for0</<2with «i(0)=l;
«2=—«i—2^+1,forO </<2with «2(0)=0;
«3=— «i— e'-1-1,for0</<2with «3(0)=1;
h=0.5;actual solutions «i(/)=cos/4-sin/— e‘4-1,«2(f)=-sin/4-cos/-eand
«3(/)=-sin/4-cos/.
d.u\=«2-«34-/,for0</<1with «i(0)=1;
«2=3r2,forO </<1with «2(0)=1;
M3=«24-e_
l,f o r0</<lwith «3(0)=— 1;
h=0.1;actual solutions «i(/)=— 0.05 /54-0.25 /44-/4-2— e~\ «2(0=t34-1,and
M3(/)=0.25 /44-/-e~'.
2. UsetheRunge -Kutta method forsystems toapproximate thesolutions ofthefollowing higher -order
differential equations andcompare theresults totheactual solutions .
a.y"— 2y'4-y=te‘— /,for0</<1with y(0)=y'(0)=0andh=0.1;actual solution
y(0=i/3e'-tel4-2e‘-/-2.
b.t2y"-2/y'4-2y=/3In/,for1</<2with y(l)=1,y'(l)=0,andh=0.1;actual solution
y(0=^+^3l n/-|/3.
c.ym4-2y"-/-2y= for0</<3with y(0)=1,y'(0)=2,y"(0)=0,andh=0.2;
actual solution y(/)= 4- — ^e-2/-I-lte‘.
d./3yw-t2y"4-3ty'-4y=5/3In/+91\for1</<2with y(l)=0,/(1)=1,/'(1)=3,
andh=0.1;actual solution y(/)=— /24-/cos(ln/)4-/sin(ln/)4-/3I n/.
3. Change theAdams Fourth -Order Predictor -Corrector method toobtain approximate solutions to
systems offirst-order equations .
4. Repeat Exercise 1using themethod developed inExercise 3.
5. Thestudy ofmathematical models forpredicting thepopulation dynamics ofcompeting species hasits
origin inindependent works published intheearly partofthiscentury byA.J.Lotka andV.Volterra .
Consider theproblem ofpredicting thepopulation oftwospecies ,oneofwhich isapredator ,whose
population attime /is*2(/),feeding ontheother ,which istheprey,whose population is*,(/).We
willassume thattheprey always hasanadequate food supply andthatitsbirth rateatanytime is
proportional tothenumber ofprey alive atthattime;thatis,birth rate (prey )isk\X\(t).Thedeath
rateoftheprey depends onboth thenumber ofprey andpredators alive atthattime.Forsimplicity ,
weassume death rate(prey)=k2xi(/)x2(/).Thebirth rateofthepredator ,ontheother hand ,depends
onitsfood supply ,X\(/),aswellasonthenumber ofpredators available forreproduction purposes .
Cop>?i£tu2012 Cc«£»fcLearn in*.AIR.(huRocncd Mayr*xbecopied ,canned ,o*daplicaied.»whole o»mpan .Doctoelectronic rifhu.*wcthird pony contcac may besoppre ^tedftem theeBook and/orcCh<xcn»l.Editorial roiew h*>
deemed CutanyMpprcucd content dee>notnuxtiily alTect theoverall leamir*experience .C'cnitapc [.camon roentt thety-ht»renxyve additional conceal atanytime i itutoeqjroi nght »rotrictionc require It222 C H A P T E R 5 Numerical Solution ofInitial -Value Problems
Forthisreason ,weassume thatthebirth rate(predator )iskyXi(0*2(0-’Thedeath rateofthepredator
willbetaken assimply proportional tothenumber ofpredators alive atthetime;thatis,death rate
(predator )=kiX 2(t).
Since xj(0andxj(0represent thechange intheprey andpredator populations ,respectively ,
with respect totime ,theproblem isexpressed bythesystem ofnonlinear differential equations
*J(0=*i*i(0-Mi(0*2(0and*2(0=Mi(0*2(0-*4*2(0.
UseRunge -Kutta oforder 4forsystems tosolve thissystem for0<t<4,assuming thattheinitial
population oftheprey is1000 andofthepredators is500 andthattheconstants arek\=3,k2=
0.002 ,*3=0.0006 .andk4=0.5.Isthere astable solution tothispopulation model ?Ifso,forwhat
values x\andxzisthesolution stable ?
6. InExercise 5weconsidered theproblem ofpredicting thepopulation inapredator -prey model .
Another problem ofthistype isconcerned with twospecies competing forthesame food supply .
Ifthenumbers ofspecies alive attime taredenoted by*i(/)and xj(/),itisoften assumed that,
although thebirth raleofeach ofthespecies issimply proportional tothenumber ofspecies alive
atthattime ,thedeath rateofeach species depends onthepopulation ofbothspecies .Wewillassume
thatthepopulation ofaparticular pairofspecies isdescribed bytheequations
=*,(/)[4-0.0003 *,(1)-0.0004 *i(r)]
and
dx]{,)=*2(r)[2-0.0002 *,(f)-0.0001 *2(1)).at
Ifitisknown thattheinitial population ofeach species is10,000,findthesolution tothissystem for
0<t<4.Isthere astable solution tothispopulation model ?Ifso,forwhat values ofx,andx2is
thesolution stable ?
5.8 StiffDifferential Equations
Stiff systems derive their name
from themotion ofspring and
mass systems thathave large
spring constants .Allthemethods forapproximating thesolution toinitial -value problems have error terms
thatinvolve ahigher derivative ofthesolution oftheequation .Ifthederivative canbe
reasonably bounded ,then themethod willhave apredictable error bound thatcanbeused
toestimate theaccuracy oftheapproximation .Even ifthederivative grows asthestepsizes
increase ,theerror canbekept inrelative control ,provided that thesolution also grows
inmagnitude .Problems frequently arise,however ,where themagnitude ofthederivative
increases ,butthesolution does not.Inthissituation ,theerror cangrow solarge that
itdominates thecalculations .Initial -value problems forwhich thisislikely tooccur are
called stifTequations andarequite common ,particularly inthestudy ofvibrations ,chemical
reactions ,andelectrical circuits .
Stiff differential equations have anexact solution with aterm oftheform e~c\where
cisalarge positive constant .This isusually only apartofthesolution ,called thetransient
solution ;themore important portion ofthesolution iscalled thesteady -state solution .A
transient portion ofastiff equation willrapidly decay tozeroastincreases ,butsince the
nthderivative ofthisterm hasmagnitude cne~c\thederivative does notdecay asquickly ,
andforlarge values ofcitcangrow very large.Inaddition ,thederivative intheerror
term isevaluated notatf,butatanumber between zero and f,sothederivative terms may
increase astincreases — andvery rapidly indeed .Fortunately ,stiffequations cangenerally
bepredicted from thephysical problem from which theequation isderived ,andwith care
theerror canbekept under control .Themanner inwhich thisisdone isconsidered inthis
section .
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