ecuaciones diferenciales Dennis-Zill

difEecrueancciioanleess

con problemas con valores
en la frontera

Octava edición

Dennis G. Zill
Warren S. Wright

OCTAVA EDICIÓN

ECUACIONES
DIFERENCIALES

con problemas con valores
en la frontera



OCTAVA EDICIÓN

ECUACIONES
DIFERENCIALES

con problemas con valores
en la frontera

DENNIS G. ZILL

Loyola Marymount University

WARREN S. WRIGHT

Loyola Marymount University
MICHAEL R. CULLEN
Antiguo miembro de la Loyola Marymount University

TRADUCCIÓN
Dra. Ana Elizabeth García Hernández
Profesor invitado UAM-Azcapotzalco

REVISIÓN TÉCNICA
Dr. Edmundo Palacios Pastrana
Universidad Iberoamericana

Australia • Brasil • Corea • España • Estados Unidos • Japón • México • Reino Unido • Singapur

Ecuaciones diferenciales © D.R. 2015 por Cengage Learning Editores, S.A. de C.V.,
con problemas con valores en la frontera una Compañía de Cengage Learning, Inc.
Octava edición Corporativo Santa Fe
Dennis G. Zill y Warren S. Wright Av. Santa Fe núm. 505, piso 12
Col. Cruz Manca, Santa Fe
Presidente de Cengage Learning C.P. 05349, México, D.F.
Latinoamérica: Cengage Learning™ es una marca registrada
Fernando Valenzuela Migoya usada bajo permiso.

Director Editorial, de Producción y de DERECHOS RESERVADOS. Ninguna parte de
Plataformas Digitales para Latinoamérica: este trabajo amparado por la Ley Federal del
Ricardo H. Rodríguez Derecho de Autor, podrá ser reproducida,
transmitida, almacenada o utilizada en
Editora de Adquisiciones para cualquier forma o por cualquier medio, ya sea
Latinoamérica: gráfico, electrónico o mecánico, incluyendo,
Claudia C. Garay Castro pero sin limitarse a lo siguiente: fotocopiado,
reproducción, escaneo, digitalización,
Gerente de Manufactura para grabación en audio, distribución en Internet,
Latinoamérica: distribución en redes de información o
Raúl D. Zendejas Espejel almacenamiento y recopilación en sistemas
de información a excepción de lo permitido
Gerente Editorial en Español para en el Capítulo III, Artículo 27 de la Ley Federal
Latinoamérica: del Derecho de Autor, sin el consentimiento
Pilar Hernández Santamarina por escrito de la Editorial.

Gerente de Proyectos Especiales: Traducido del libro Differential Equations with
Luciana Rabuffetti Boundary-Value Problems, Eighth Edition
Publicado en inglés por Brooks/Cole, Cengage
Coordinador de Manufactura: Learning © 2013
Rafael Pérez González
Datos para catalogación bibliográfica:
Editor: Zill, Dennis G. y Warren S. Wright
Omegar Martínez Ecuaciones diferenciales con problemas
con valores en la frontera, octava edición
Diseño de portada: ISBN: 978-607-519-444-8
Anneli Daniela Torres Arroyo
Visite nuestro sitio en:
Imagen de portada: http://latinoamerica.cengage.com
Space, © Rolffimages / Dreamstime.com

Composición tipográfica:
Aurora Esperanza López López

Impreso en México
1 2 3 4 5 6 7 17 16 15 14

CONTENIDO 1

Prefacio xi
Proyectos P-1

1 INTRODUCCIÓN A LAS ECUACIONES DIFERENCIALES

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2 ECUACIONES DIFERENCIALES DE PRIMER ORDEN 34
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3 MODELADO CON ECUACIONES DIFERENCIALES DE PRIMER ORDEN

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v

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4 ECUACIONES DIFERENCIALES DE ORDEN SUPERIOR 113

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6 SOLUCIONES EN SERIES DE ECUACIONES LINEALES 225

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CONTENIDO l vii

7 LA TRANSFORMADA DE LAPLACE 265

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8 SISTEMAS DE ECUACIONES DIFERENCIALES LINEALES DE PRIMER ORDEN 317

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9 SOLUCIONES NUMÉRICAS DE ECUACIONES DIFERENCIALES ORDINARIAS 353

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viii l CONTENIDO 376

10 SISTEMAS AUTÓNOMOS PLANOS

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12 PROBLEMAS CON VALORES EN LA FRONTERA EN COORDENADAS RECTANGULARES 445

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CONTENIDO l ix

13 PROBLEMAS CON VALORES EN LA FRONTERA EN OTROS SISTEMAS COORDENADOS 483

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14 TRANSFORMADA INTEGRAL 500

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15 SOLUCIONES NUMÉRICAS DE ECUACIONES DIFERENCIALES PARCIALES 524

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,YDQ .UDPHU University of Maryland, Baltimore County
7RP /D)DUR Gustavus Adolphus College
-R *DVFRLJQH Fisheries Consultant
& - .QLFNHUERFNHU Sensis Corporation
.HYLQ &RRSHU Washington State University
*LOEHUW 1 /HZLV Michigan Technological University
0LFKDHO 2OLQLFN Middlebury College

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REVISORES DE EDICIONES ANTERIORES

:LOOLDP $WKHUWRQ Cleveland State University
3KLOLS %DFRQ University of Florida
%UXFH %D\O\ University of Arizona
:LOOLDP + %H\HU University of Akron
5 * %UDGVKDZ Clarkson College
'HDQ 5 %URZQ Youngstown State University
'DYLG %XFKWKDO University of Akron
1JX\HQ 3 &DF University of Iowa
7 &KRZ California State University, Sacramento

xiv l PREFACIO

'RPLQLF 3 &OHPHQFH North Carolina Agricultural and Technical State

University

3DVTXDOH &RQGR University of Massachusetts, Lowell
9LQFHQW &RQQROO\ Worcester Polytechnic Institute
3KLOLS 6 &URRNH Vanderbilt University
%UXFH ( 'DYLV St. Louis Community College at Florissant Valley
3DXO : 'DYLV Worcester Polytechnic Institute
5LFKDUG $ 'L'LR La Salle University
-DPHV 'UDSHU University of Florida
-DPHV 0 (GPRQGVRQ Santa Barbara City College
-RKQ + (OOLVRQ Grove City College
5D\PRQG )DEHF Louisiana State University
'RQQD )DUULRU University of Tulsa
5REHUW ( )HQQHOO Clemson University
: ( )LW]JLEERQ University of Houston
+DUYH\ - )OHWFKHU Brigham Young University
3DXO - *RUPOH\ Villanova
/D\DFKL +DGML University of Alabama
5XEHQ +D\UDSHW\DQ Kettering University
7HUU\ +HUGPDQ Virginia Polytechnic Institute and State University
=G]LVODZ -DFNLHZLF] Arizona State University
6 . -DLQ Ohio University
$QWKRQ\ - -RKQ Southeastern Massachusetts University
'DYLG & -RKQVRQ University of Kentucky, Lexington
+DUU\ / -RKQVRQ V.P.I & S.U.
.HQQHWK 5 -RKQVRQ North Dakota State University
-RVHSK .D]LPLU East Los Angeles College
- .HHQHU University of Arizona
6WHYH % .KOLHI Tennessee Technological University (retirado)
& - .QLFNHUERFNHU Sensis Corporation
&DUORQ $ .UDQW] Kean College of New Jersey
7KRPDV * .XG]PD University of Lowell
$OH[DQGUD .XUHSD North Carolina A&T State University
* ( /DWWD University of Virginia
&HFHOLD /DXULH University of Alabama
-DPHV 5 0F.LQQH\ California Polytechnic State University
-DPHV / 0HHN University of Arkansas
*DU\ + 0HLVWHUV University of Nebraska, Lincoln
6WHSKHQ - 0HUULOO Marquette University
9LYLHQ 0LOOHU Mississippi State University
*HUDOG 0XHOOHU Columbus State Community College
3KLOLS 6 0XOU\ Colgate University
& - 1HXJHEDXHU Purdue University
7\UH $ 1HZWRQ Washington State University
%ULDQ 0 2¶&RQQRU Tennessee Technological University
- . 2GGVRQ University of California, Riverside
&DURO 6 2¶'HOO Ohio Northern University
$ 3HUHVVLQL University of Illinois, Urbana, Champaign
- 3HUU\PDQ University of Texas at Arlington
-RVHSK + 3KLOOLSV Sacramento City College
-DFHN 3ROHZF]DN California State University Northridge
1DQF\ - 3R[RQ California State University, Sacramento
5REHUW 3UXLWW San Jose State University
. 5DJHU Metropolitan State College
) % 5HLV Northeastern University

PREFACIO l xv

%ULDQ 5RGULJXHV California State Polytechnic University
7RP 5RH South Dakota State University
.LPPR , 5RVHQWKDO Union College
%DUEDUD 6KDEHOO California Polytechnic State University
6HHQLWK 6LYDVXQGDUDP Embry-Riddle Aeronautical University
'RQ ( 6RDVK Hillsborough Community College
) : 6WDOODUG Georgia Institute of Technology
*UHJRU\ 6WHLQ The Cooper Union
0 % 7DPEXUUR Georgia Institute of Technology
3DWULFN :DUG Illinois Central College
-LDQSLQJ =KX University of Akron
-DQ =LMOVWUD Middle Tennessee State University
-D\ =LPPHUPDQ Towson University

REVISORES DE LAS EDICIONES ACTUALES

%HUQDUG %URRNV Rochester Institute of Technology
$OOHQ %URZQ Wabash Valley College
+HOPXW .QDXVW The University of Texas at El Paso
0XODWX /HPPD Savannah State University
*HRUJH 0RVV Union University
0DUWLQ 1DNDVKLPD California State Polytechnic University, Pomona
%UXFH 2¶1HLOO Milwaukee School of Engineering

'HQQLV * =LOO
:DUUHQ 6 :ULJKW

Los Ángeles



PROYECTO
PARA LA SECCIÓN 3.1

¿Invariablemente el SIDA es

una enfermedad fatal? por Ivan Kramer

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P-1

P-2 l PROYECTO 3.1 ¿INVARIABLEMENTE EL SIDA ES UNA ENFERMEDAD FATAL?

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PROYECTO 3.1 ¿INVARIABLEMENTE EL SIDA ES UNA ENFERMEDAD FATAL? l P-3

1.0 Fracción de supervivencia

Ajuste del modelo de dos parámetros

0.8

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0 48 80 112 144 176 208 240 272
_16 16 Tiempo de supervivencia t(w)

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Dinámica de población de lobos

por C. J. Knickerbocker

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PROYECTO 3.3 DINÁMICA DE POBLACIÓN DE LOBOS l P-9

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e1 := diff(e(t),t)-0.0325*e(t) + 0.8*e(t)*w(t) :

e2 := diff(w(t),t)+0.6*w(t) - 0.05*e(t)*w(t) :

sys := {e1,e2} :

ic := {e(0)=18.0,w(0)=0.021} :

ivp := sys union ic :

H:= dsolve(ivp,{e(t),w(t)},numeric) :

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0 0
1995 1997 1999 2001 2003 2005 2007 2009 1995 1997 1999 2001 2003 2005 2007 2009

Año Año
Población de alces
Población de lobos

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18000 180
16000 Lobos 160

Poblaciones de alces y lobos 14000 140

12000 120

10000 100

8000 80

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0 0
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ACERCA DEL AUTOR

C. J. Knickerbocker
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Puente PARA LA SECCIÓN 5.1
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Salto en bungee
Bungee
100 pies por Kevin Cooper

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REFERENCIAS

1. /HZLV * 1 ³7DFRPD 1DUURZV 6XVSHQVLRQ %ULGJH &ROODSVH´ HQ 'HQQLV * =LOO A First Course
in Differential Equations %RVWRQ 3:6 .HQW SS

2. %UDXQ 0 Differential Equations and Their Applications SS 1XHYD <RUN 6SULQJHU


3. $PPDQ 2 + 7 YRQ .iUPiQ \ * % :RRGUXII The Failure of the Tacoma Narrows Bridge
:DVKLQJWRQ ' & )HGHUDO :RUNV $JHQF\

4. /D]HU $ & \ 3 - 0F.HQQD ³/DUJH DPSOLWXGH SHULRGLF RVFLOODWLRQV LQ VXVSHQVLRQ EULGJHV 6RPH
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XVWHG PXOWLSOLFD OD PDWUL] K SRU HV GHFLU KDFH DO HGL¿FLR PiV UtJLGR " ¢3RU FXiQWR
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4. 5HVXHOYD HO SUREOHPD GHO WHUUHPRWR SDUD HO HGL¿FLR GH WUHV SLVRV GHO SUREOHPD

MX'' ϭ KX ϩ F t

donde F t ϭ G FRVȖW G ϭ EB B ϭ > @T E ϭ OLEUDV HV OD DPSOLWXG GH OD
IXHU]D GHO WHUUHPRWR TXH DFW~D DO QLYHO GHO VXHOR \ Ȗ ϭ HV OD IUHFXHQFLD GHO WHUUHPRWR
XQD IUHFXHQFLD WtSLFD GH WHUUHPRWR 9HD OD VHFFLyQ SDUD HO PpWRGR PDWULFLDO GH UHVR-
OXFLyQ GH HFXDFLRQHV GLIHUHQFLDOHV QR KRPRJpQHDV 8WLOLFH FRQGLFLRQHV LQLFLDOHV SDUD XQ
HGL¿FLR HQ UHSRVR

PROYECTO
PARA LA SECCIÓN 8.3

Modelado de carreras armamentistas

por Michael Olinick

(Q ORV ~OWLPRV FLHQ DxRV VH KDQ YLVWR QXPHURVDV FDUUHUDV DUPDPHQWLVWDV SHOLJURVDV

GHVHVWDELOL]DGRUDV \ FRVWRVDV (O HVWDOOLGR GH OD 3ULPHUD *XHUUD 0XQGLDO FXOPLQy FRQ XQD

UiSLGD DFXPXODFLyQ GH DUPDPHQWRV HQWUH ODV SRWHQFLDV HXURSHDV ULYDOHV +XER XQD DFXPX-

ODFLyQ VLPLODU PXWXD GH DUPDV FRQYHQFLRQDOHV MXVWR DQWHV GH OD 6HJXQGD *XHUUD 0XQGL-

DO /RV (VWDGRV 8QLGRV \ OD 8QLyQ 6RYLpWLFD HQWDEODURQ XQD FRVWRVD FDUUHUD DUPDPHQWLVWD

QXFOHDU GXUDQWH ORV FXDUHQWD DxRV GH OD *XHUUD )UtD (O DFRSLR GH DUPDV PRUWDOHV HV FDGD

YH] PiV FRP~Q KR\ HQ GtD HQ PXFKDV SDUWHV GHO PXQGR LQFOX\HQGR HO 0HGLR 2ULHQWH HO

VXEFRQWLQHQWH LQGLR \ OD SHQtQVXOD GH &RUHD

$UPDV \ PXQLFLRQHV UHFXSHUDGDV (O PHWHRUyORJR \ HGXFDGRU EULWiQLFR /HZLV ) 5LFKDUGVRQ GHVDUUROOy YDUL-

GXUDQWH ODV RSHUDFLRQHV PLOLWDUHV FRQWUD RV PRGHORV PDWHPiWLFRV SDUD DQDOL]DU OD GLQiPLFD GH ODV FDUUHUDV DUPDPHQWLVWDV OD HYR-
ORV PLOLWDQWHV WDOLEDQHV HQ :D]LULVWiQ
GHO 6XU HQ RFWXEUH GH OXFLyQ HQ HO WLHPSR GHO SURFHVR GH LQWHUDFFLyQ HQWUH ORV SDtVHV HQ VX DGTXLVLFLyQ GH DUPDV
/RV PRGHORV GH ODV FDUUHUDV DUPDPHQWLVWDV JHQHUDOPHQWH VXSRQHQ TXH FDGD QDFLyQ DMXVWD

VX DFXPXODFLyQ GH DUPDV GH DOJXQD PDQHUD TXH GHSHQGH GHO WDPDxR GH VXV SURSLDV UHVHUYDV

\ GHO QLYHO GH DUPDPHQWR GH ODV RWUDV QDFLRQHV

(O PRGHOR SULPDULR GH 5LFKDUGVRQ GH XQD FDUUHUD DUPDPHQWLVWD GH GRV SDtVHV VH EDVD

en el miedo mutuo XQD QDFLyQ HV HVWLPXODGD D DXPHQWDU VX DUPDPHQWR FRQ XQD YHORFLGDG

SURSRUFLRQDO DO QLYHO GH JDVWRV HQ DUPDPHQWR GH VX ULYDO (O PRGHOR GH 5LFKDUGVRQ WRPD HQ

FXHQWD ODV UHVWULFFLRQHV LQWHUQDV GHQWUR GH XQD QDFLyQ TXH IUHQDQ OD DFXPXODFLyQ GH DUPDPHQ-

WRV (QWUH PiV JDVWH XQD QDFLyQ HQ DUPDV PiV GLItFLO HV KDFHU LQFUHPHQWRV PD\RUHV SRUTXH

VH YXHOYH FDGD YH] PiV GLItFLO GHVYLDU UHFXUVRV GH ODV QHFHVLGDGHV EiVLFDV GH OD VRFLHGDG

FRPR DOLPHQWDFLyQ \ YLYLHQGD D ODV DUPDV 5LFKDUGVRQ WDPELpQ FRQWHPSOy HQ VX PRGHOR

RWURV IDFWRUHV TXH FRQGXFHQ R IUHQDQ XQD FDUUHUD DUPDPHQWLVWD TXH VRQ LQGHSHQGLHQWHV GH ORV

QLYHOHV GH JDVWRV HQ DUPDPHQWRV /D HVWUXFWXUD PDWHPiWLFD GH HVWH PRGHOR HV XQ VLVWHPD GH

HFXDFLRQHV GLIHUHQFLDOHV GH SULPHU RUGHQ HQOD]DGDV 6L x y y UHSUHVHQWDQ OD FDQWLGDG GH ULTXH-

]D TXH VH JDVWD HQ DUPDV SRU ODV GRV QDFLRQHV DO WLHPSR t HQWRQFHV HO PRGHOR WRPD OD IRUPD

dx
ay mx r

dt
dy

bx ny s
dt

donde a b m y n VRQ FRQVWDQWHV SRVLWLYDV PLHQWUDV TXH r y s VRQ FRQVWDQWHV TXH SXHGHQ VHU
SRVLWLYDV R QHJDWLYDV /DV FRQVWDQWHV a y b PLGHQ HO PLHGR PXWXR ODV FRQVWDQWHV m y n repre-
VHQWDQ IDFWRUHV GH SURSRUFLRQDOLGDG SDUD ORV ³IUHQRV LQWHUQRV´ D QXHYRV DXPHQWRV GH DUPDV
9DORUHV SRVLWLYRV GH r y s FRUUHVSRQGHQ D IDFWRUHV VXE\DFHQWHV GH PDOD YROXQWDG R GHVFRQ¿DQ-
]D TXH SHUVLVWLUtDQ DXQ VL ORV JDVWRV HQ DUPDV VH UHGXMHUDQ D FHUR /RV YDORUHV QHJDWLYRV SDUD r
y s LQGLFDQ XQD FRQWULEXFLyQ EDVDGD HQ OD EXHQD YROXQWDG

(O FRPSRUWDPLHQWR GLQiPLFR GH HVWH VLVWHPD GH HFXDFLRQHV GLIHUHQFLDOHV GHSHQGH GH ORV
WDPDxRV UHODWLYRV GH ab y mn MXQWR FRQ ORV VLJQRV GH r y s $XQTXH HO PRGHOR HV UHODWLYDPHQWH
VLPSOH SHUPLWH FRQVLGHUDU GLIHUHQWHV UHVXOWDGRV D ODUJR SOD]R (V SRVLEOH TXH GRV QDFLRQHV VH
SXGLHUDQ PRYHU VLPXOWiQHDPHQWH KDFLD HO GHVDUPH PXWXR FRQ x y y DFHUFiQGRVH D FHUR 8Q
FtUFXOR YLFLRVR HV DXPHQWDU LOLPLWDGDPHQWH D x y y HV RWUR HVFHQDULR SRVLEOH 8QD WHUFHUD
SRVLELOLGDG HV TXH ORV JDVWRV HQ DUPDV VH DFHUTXHQ DVLQWyWLFDPHQWH D XQ SXQWR HVWDEOH
x* y* LQGHSHQGLHQWHPHQWH GHO QLYHO LQLFLDO GH JDVWRV GH DUPDPHQWRV (Q RWURV FDVRV HO
UHVXOWDGR ¿QDO GHSHQGH GHO SXQWR GH SDUWLGD /D ¿JXUD PXHVWUD XQD SRVLEOH VLWXDFLyQ FRQ
FXDWUR GLIHUHQWHV QLYHOHV LQLFLDOHV FDGD XQR GH HOORV FRQGXFH D XQ ³UHVXOWDGR HVWDEOH´ OD
LQWHUFHVLyQ GH ODV FHURFOLQDV dx΋dt ϭ \ dy΋dt ϭ

P-24

PROYECTO 8.3 MODELADO DE CARRERAS ARMAMENTISTAS l P-25

y $XQTXH ODV FDUUHUDV DUPDPHQWLVWDV GHO ³PXQGR UHDO´ UDUD YH] FRLQFLGHQ H[-
dx/dt = 0
DFWDPHQWH FRQ HO PRGHOR GH 5LFKDUGVRQ VX SULPHU WUDEDMR KD FRQGXFLGR
6
D PXFKDV DSOLFDFLRQHV IUXFWtIHUDV GH PRGHORV GH HFXDFLRQHV GLIHUHQFLDOHV D
5
dy/dt = 0 SUREOHPDV HQ UHODFLRQHV LQWHUQDFLRQDOHV \ FLHQFLDV SROtWLFDV &RPR DQRWDQ ORV
4
GRV LQYHVWLJDGRUHV SULQFLSDOHV HQ OD UHIHUHQFLD >3@ ³HO PRGHOR GH FDUUHUD DU-
3
PDPHQWLVWD GH 5LFKDUGVRQ FRQVWLWX\H XQR GH ORV PRGHORV PiV LPSRUWDQWHV GH
2
ORV IHQyPHQRV GH FDUUHUD DUPDPHQWLVWD \ DO PLVPR WLHPSR XQR GH ORV PRGHORV
1
IRUPDOHV TXH PiV LQÀX\HQ HQ ODV SXEOLFDFLRQHV GH UHODFLRQHV LQWHUQDFLRQDOHV´
0 1234
/DV FDUUHUDV DUPDPHQWLVWDV QR VH OLPLWDQ D OD LQWHUDFFLyQ GH SDtVHV
FIGURA 1 *DVWRV
DFHUFiQGRVH D XQ SXQWR HVWDEOH 3XHGHQ WHQHU OXJDU HQWUH XQ JRELHUQR \ XQ JUXSR WHUURULVWD GH SDUDPLOLWDUHV

GHQWUR GH VXV IURQWHUDV FRPR SRU HMHPSOR ORV WLJUHV WDPLOHV HQ 6UL /DQ-

ND 6HQGHUR /XPLQRVR HQ 3HU~ R HO WDOLEiQ HQ $IJDQLVWiQ 7DPELpQ VH KDQ

REVHUYDGR IHQyPHQRV DUPDPHQWLVWDV HQWUH EDQGDV ULYDOHV XUEDQDV \ HQWUH

x RUJDQLVPRV GH UHSUHVLyQ \ FULPHQ RUJDQL]DGR
56 /DV ³DUPDV´ QR VRQ QHFHVDULDPHQWH DUPDV /DV XQLYHUVLGDGHV KDQ SDU-

WLFLSDGR HQ ³VHUYLFLRV DUPDPHQWLVWDV´ D PHQXGR GHGLFDQGR PLOORQHV GH GyODUHV HQ GRUPLWRULRV

PiV OXMRVRV LQVWDODFLRQHV GHSRUWLYDV GH YDQJXDUGLD RSFLRQHV JDVWURQyPLFDV VLEDULWDV \ VLPLODUHV

SDUD VHU PiV FRPSHWLWLYRV \ DWUDHU DVt LQVFULSFLRQHV GH HVWXGLDQWHV /RV ELyORJRV KDQ LGHQWL¿FDGR

OD SRVLELOLGDG GH TXH H[LVWDQ FDUUHUDV DUPDPHQWLVWDV HYROXFLRQDULDV HQWUH \ GHQWUR GH DOJXQDV HV-

SHFLHV GH WDO PRGR TXH XQ DGDSWDFLyQ GHQWUR GH XQ OLQDMH SXHGH RFXUULU SRU OD SUHVLyQ VHOHFWLYD GH

RWUR OLQDMH GDQGR OXJDU D XQD FRQWUD DGDSWDFLyQ 'H PDQHUD PiV JHQHUDO ODV VXSRVLFLRQHV SUHVHQ-

WDQ XQ WLSR GH PRGHOR GH 5LFKDUGVRQ TXH WDPELpQ FDUDFWHUL]D PXFKDV FRPSHWLFLRQHV HQ ODV TXH

FDGD ODGR SHUFLEH OD QHFHVLGDG GH DGHODQWDUVH D RWUR PXWXDPHQWH HQ DOJXQD PHGLGD LPSRUWDQWH

PROBLEMAS RELACIONADOS

1. a) 0 HGLDQWH OD VXVWLWXFLyQ GH ODV VROXFLRQHV SURSXHVWDV D ODV HFXDFLRQHV GLIHUHQFLDOHV
GHPXHVWUH TXH OD VROXFLyQ GHO PRGHOR SDUWLFXODU DUPDPHQWLVWD GH 5LFKDUGVRQ

dx
y 3x 3

dt

dy
2x 4y 8

dt

FRQ OD FRQGLFLyQ LQLFLDO x ϭ y ϭ HV

x(t) 32 2t 2 5t 2
e e
33

y(t) 32 2t 4 5t 3
e e
33

¢&XiO HV HO FRPSRUWDPLHQWR D ODUJR SOD]R GH HVWD FDUUHUD DUPDPHQWLVWD"

b) 3 DUD HO PRGHOR GH FDUUHUD DUPDPHQWLVWD GH 5LFKDUGVRQ VL D FRQ FRQGLFLRQHV LQLFLDOHV
arbitrarias x ϭ A y ϭ B GHPXHVWUH TXH OD VROXFLyQ HVWi GDGD SRU

x(t) Ce 5t De 2t 2 donde C (A B 1) 3
D (2A B 7) 3
y(t) 2Ce 5t De 2t 3

'HPXHVWUH TXH HVWH UHVXOWDGR LPSOLFD TXH HO FRPSRUWDPLHQWR FXDOLWDWLYR D ODUJR
SOD]R GH HVWD FDUUHUD DUPDPHQWLVWD HV HO PLVPR x t → y t → VLQ LPSRUWDU
FXiOHV VRQ ORV YDORUHV LQLFLDOHV GH x y y

2. (O FRPSRUWDPLHQWR FXDOLWDWLYR D ODUJR SOD]R GH XQ PRGHOR GH FDUUHUD DUPDPHQWLVWD GH
5LFKDUGVRQ SXHGH HQ DOJXQRV FDVRV GHSHQGHU GH ODV FRQGLFLRQHV LQLFLDOHV &RQVLGHUHPRV
SRU HMHPSOR HO VLVWHPD
dx
3y 2x 10
dt

dy
4x 3y 10

dt

P-26 l PROYECTO 8.3 MODELADO DE CARRERAS ARMAMENTISTAS

3DUD FDGD XQD GH ODV FRQGLFLRQHV LQLFLDOHV GDGDV D FRQWLQXDFLyQ FRPSUXHEH TXH OD VROX-
FLyQ SURSXHVWD IXQFLRQD \ DQDOLFH HO FRPSRUWDPLHQWR D ODUJR SOD]R
a) x(0) 1, y(0) 1 : x(t) 10 9et, y(t) 10 9et
b) x(0) 1, y(0) 22 : x(t) 10 9e 6t, y(t) 10 12e 6t
c) x(0) 1, y(0) 29 : x(t) 12e 6t 3et 10, y(t) 16e 6t 3et 10
d) x(0) 10, y(0) 10 : x(t) 10, y(t) 10 para todo t

3. a) & RPR XQD SRVLEOH DOWHUQDWLYD DO PRGHOR GH 5LFKDUGVRQ FRQVLGHUH XQ PRGHOR GH
ajuste de inventarios /D VXSRVLFLyQ DTXt HV TXH FDGD SDtV HVWDEOHFH SRU Vt PLVPR XQ
QLYHO GHVHDGR GH JDVWRV HQ DUPDPHQWR \ OXHJR FDPELD VX LQYHQWDULR GH DUPDV SUR-
SRUFLRQDOPHQWH D OD GLIHUHQFLD HQWUH VX QLYHO DFWXDO \ VX QLYHO GHVHDGR 'HPXHVWUH
TXH HVWD KLSyWHVLV VH SXHGH UHSUHVHQWDU SRU HO VLVWHPD GH HFXDFLRQHV GLIHUHQFLDOHV

dx
a(x* x)

dt

dx
b(y* y)

dt

donde x* y y* VRQ ORV QLYHOHV GHVHDGRV GH FRQVWDQWHV \ a b VRQ FRQVWDQWHV SRVLWL-
YDV ¢&yPR HYROXFLRQDQ x y y FRQ HO WLHPSR EDMR HVH PRGHOR"

b) *HQHUDOLFH HO PRGHOR GH DMXVWH GH H[LVWHQFLD GHO LQFLVR D D XQ QLYHO PiV UHDOLVWD
GRQGH HO QLYHO GHVHDGR SDUD FDGD SDtV GHSHQGD GH ORV QLYHOHV GH DPERV SDtVHV (Q
SDUWLFXODU VXSRQJDPRV TXH x* WLHQH OD IRUPD x* ϭ c ϩ dy donde c y d VRQ FRQVWDQWHV
SRVLWLYDV \ TXH y* WLHQH XQD IRUPD VLPLODU 'HPXHVWUH TXH EDMR HVWDV VXSRVLFLRQHV
HO PRGHOR GH DMXVWH GH H[LVWHQFLD HV HTXLYDOHQWH D XQ PRGHOR GH 5LFKDUGVRQ

4. ([WLHQGD HO PRGHOR GH 5LFKDUGVRQ D WUHV QDFLRQHV GHGXFLHQGR XQ VLVWHPD GH HFXDFLRQHV
GLIHUHQFLDOHV OLQHDOHV VL ORV WUHV VRQ PXWXDPHQWH WHPHURVRV FDGD XQR HV HVWLPXODGR SDUD
DUPDUVH SRU ORV JDVWRV GH ORV RWURV GRV ¢&yPR FDPELDUtDQ ODV HFXDFLRQHV VL GRV GH ODV QD-
FLRQHV VRQ DOLDGDV FHUFDQDV QR DPHQD]DGDV SRU OD DFXPXODFLyQ GH ODV DUPDV GHO RWUR SHUR
WHPHURVDV GH ORV DUPDPHQWRV GH OD WHUFHUD" ,QYHVWLJXH HO FRPSRUWDPLHQWR D ODUJR SOD]R GH
HVWDV FDUUHUDV DUPDPHQWLVWDV

5. (Q HO PXQGR UHDO XQD FDUUHUD GH DUPDPHQWRV QR DFRWDGD HV LPSRVLEOH \D TXH KD\ XQ OtPLWH
D OD FDQWLGDG TXH FXDOTXLHU SDtV SXHGH JDVWDU HQ DUPDV SRU HMHPSOR HO SURGXFWR QDFLRQDO
EUXWR PHQRV FLHUWD FDQWLGDG SDUD OD VXSHUYLYHQFLD 0RGL¿TXH HO PRGHOR GH 5LFKDUGVRQ
SDUD LQFRUSRUDU HVWD LGHD \ DQDOLFH OD GLQiPLFD GH XQD FDUUHUD DUPDPHQWLVWD TXH VH ULJH SRU
HVWDV QXHYDV HFXDFLRQHV GLIHUHQFLDOHV

REFERENCIAS

1. 5LFKDUGVRQ /HZLV ) Arms and Insecurity: A Mathematical Study of the Cause and Origins of
War 3LWWVEXUJK %R[ZRRG 3UHVV

2. 2OLQLFN 0LFKDHO An Introduction to Mathematical Models in the Social and Life Sciences
5HDGLQJ $GGLVRQ :HVOH\

3. ,QWULOLJDWRU 0LFKDHO ' \ 'DJREHUW / %ULWR ³5LFKDUGVRQLDQ $UPV 5DFH 0RGHOV´ HQ 0DQXV ,
0LGODUVN\ HG Handbook of War Studies %RVWRQ 8QZLQ +\PDQ

ACERCA DEL AUTOR

'HVSXpV GH REWHQHU OD OLFHQFLDWXUD HQ PDWHPiWLFDV \ ¿ORVRItD HQ OD 8QLYHUVLGDG GH 0LFKLJDQ
\ OD PDHVWUtD \ HO GRFWRUDGR HQ OD 8QLYHUVLGDG GH :LVFRQVLQ 0DGLVRQ Michael Olinick se
PXGy GHO PHGLR RHVWH GH (VWDGRV 8QLGRV D 1XHYD ,QJODWHUUD $Kt VH XQLy D OD IDFXOWDG GHO
0LGGOHEXU\ &ROOHJH HQ FRPR SURIHVRU GH PDWHPiWLFDV (O 'U 2OLQLFN WLHQH SXHVWRV
GH SURIHVRU YLVLWDQWH HQ OD 8QLYHUVLW\ &ROOHJH 1DLUREL OD :HVOH\DQ 8QLYHUVLW\ OD 8QLYHU-
VLGDG GH &DOLIRUQLD HQ %HUNHOH\ \ OD 8QLYHUVLGDG GH /DQFDVWHU HQ *UDQ %UHWDxD (V DXWRU
R FRDXWRU GH YDULRV OLEURV GH FiOFXOR GH XQD \ PXFKDV YDULDEOHV GH PRGHODGR PDWHPiWLFR
SUREDELOLGDG \ WRSRORJtD $FWXDOPHQWH HVWi GHVDUUROODQGR XQ QXHYR OLEUR VREUH PRGHORV
PDWHPiWLFRV HQ KXPDQLGDGHV FLHQFLDV VRFLDOHV \ FLHQFLDV GH OD YLGD

1 INTRODUCCIÓN A LAS
ECUACIONES DIFERENCIALES

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REPASO DEL CAPÍTULO 1

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VHUi UHVROYHU HFXDFLRQHV GLIHUHQFLDOHV GHO WLSR yЉ ϩ 2yЈ ϩ y ϭ SDUD OD IXQFLyQ
LQFyJQLWD y ϭ ␾(x).

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