´ INTRODUCCION A LAS ECUACIONES DIFERENCIALES Jorge Adelmo Hern´ andez andez Pardo* Especialista en Matem´ atica atica Avanzada Universidad Nacional de Colombia Profesor Asistente de la Universidad Distrital “Franc “Francisco isco Jos´e de Caldas” Caldas ”
Rodrigo Rinc´ on on Zarta** Especialista en Matem´ atica atica Avanzada Universidad Nacional de Colombia Profesor Asistente de la Universidad Distrital “Franc “Francisco isco Jos´e de Caldas” Caldas ”
Noviembre de 2007
* email ** email
[email protected].
[email protected]
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Prefacio El texto recopila la experiencia docente acumulada por cada uno de los autores durante m´as a s de dos a˜nos nos orientando la c´atedra atedra “Introducci´on o n a las Ecuaciones Diferenciales”. Sabemos que responde respo nde a las necesidades acad´emiemicas de las facultades tecnol´ogicas ogicas y faculta f acultades des de ingenier´ i ngenier´ıa. ıa. El pronto desarrollo de “La Transformada de Laplace ”, se hace con el animo de que muchas de las ecuaciones diferenciales con condiciones iniciales se puedan solucionar solucionar utilizando utilizando la transformad transformadaa enunciada enunciada.. El texto contiene el desarrollo te´orico orico b´asico asico para un curso de ecuaciones diferenciales cl´asico asico y puede ser, si as´ as´ı se requiere, requiere, complementado complementado por algunos de los textos de la bibliograf´ bibliograf´ıa rese˜ nada al final de este libro. nada Por otra parte, los elevados costos de los textos tradicionales impiden que muchos de nuestros estudiantes tenga la posibilidad de adquirir uno de dichos ejemplares. Este libro pretende, entre otros, subsanar esta situaci´on. on. El libro tiene tres cap´ cap´ıtulos, ıtulos, el primero primero de estos hace un tratamien tratamiento to detallado de las integrales impropias (sin demostraciones), para aquellos estudiantes que no tengan conocimientos frescos de estos temas y por tanto podr po dr´´ıan ser tomados tomado s unicamente u ´ nicamente como material de consulta. Dicho cap´ cap´ıtulo est´a hecho m´as as o menos a la manera del libro [1]. En el segundo segundo cap´ cap´ıtulo, titulado “Transf “Transformad ormadaa de Laplace”, Laplace”, se hace un desarrollo m´as as o menos profundo de la transformada de Laplace para funi
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ii
CAP´ ITULO 0. PREFACIO
ciones causales, y que de acuerdo con el orden establecido por el libro, es una aplicaci´on on inmediata de las integrales impropias de primera clase. Adem´as, se desarrollan algunas aplicaciones en la soluci´on on de ecuaciones diferenciales, ecuaciones integrales y ecuaciones integro-diferenciales. El tercer cap´ cap´ıtulo contiene un estudio r´apido apido de las ecuaciones ecuaciones diferenciales diferenciales ordinarias de primer orden y orden superior con aplicaciones, el desarrollo es cl´asico asico a la manera de [2]. Adem´ as as contiene contiene una peque˜ na na pero util u ´ til introducci´on on a las sucesiones y series y tambi´ en en una tabla de transformadas de Laplace. Este libro es el resultado de dos ediciones de Notas de Clase publicadas por el Fondo de Publicaciones Universidad Distrital entre los a˜nos n os 2004 y 2006 con un tiraje de 300 ejemplares por cada edici´on. on. La excelente acogida nos motiv´o a complementar el contenido y proponerle a las directivas de la Universidad Distrital la publicaci´on on como libro de texto. Despu´ es es de ser evaluado evaluado positivamen positivamente te por pares externos externos fue aprobada aprobada por el comit´ comit´e de publicaciones y de esta forma se hace realidad nuestra pretensi´on. Los autores agradecemos las sugerencias y observaciones que nos hicieron tanto colegas como estudiantes sobre las Notas de Clase y tambi´ en en agradecemos cualquier sugerencia que los lectores hagan a este texto y las tendr´an en cuenta para mejorar una futura edici´on. on.
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Jorge Adelmo A mi nieto Juan Daniel, a mi hija Sandra Rodrigo, a mis hijos Camila Andrea, Diego Armando
iii
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iv
CAP´ ITULO 0.
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´ Indice general Prefacio
I
III
1. INTEGRALES IMPROPIAS 1.1 1.1. Integrales impropi opias de primera clase 1.2 1.2. Integrales impropi opias de segun gunda clase 1.2.1. Caso 1: . . . . . . . . . . . . . 1.2.2. Caso 2: . . . . . . . . . . . . . 1.3. Criterios de convergencia . . . . . . . 1.4. Funci´ unci´ on Gama . . . . . . . . . . . . .
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2. TRANSFORMADA DE LAPLACE 2.1. Introd Introducc ucci´ i´ on . . . . . . . . . . . . . . . . . . 2.2. 2.2. Prop Propie ieda dade dess de la tran transf sfor orma mada da de La Lapl plac acee 2.3. Traslaciones y derivadas . . . . . . . . . . . 2.4. Convoluci´ Convoluci´ on . . . . . . . . . . . . . . . . . . 2.5. Interpre Interpretaci´ taci´ on on de la convoluci´on . . . . . . . 2.6. Transformada ransformada de Laplace Laplace de la convoluci´ convoluci´ on 2.7. T. de Laplac Laplacee de una funci´ funci´ on on peri´odica . . .
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vi
´ INDICE GENERAL
3.4. Variables separables . . . . . . . . . . . . . . . . . . . . . . . 61 3.5. Homog´eneas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.6. 3.6. Ecua Ecuaci cion ones es dife diferrenci encial ales es exact xactas as . . . . . . . . . . . . . . . . . 66 3.6.1. M´etodo etodo de soluci´ on on de una ecuaci´on exacta . . . . . . 67 3.7. Lineales de primer orden . . . . . . . . . . . . . . . . . . . . . 72 3.8. Problemas Problemas de aplicaci´ aplicaci´ on . . . . . . . . . . . . . . . . . . . . . 76 Ley de enfriam amiiento de Newton . . . . . . . . . . . . . . . . . 78 3.9. 3.9. Ecua Ecuaci cion ones es line lineal ales es homo homogg´enea e neass . . . . . . . . . . . . . . . . . 86 3.9.1. 3.9.1. Ecuaci Ecuacione oness no-hom no-homog´ og´ eneas eneas de segund segundoo orden orden . . . . . 89 3.9.2. Ecuaciones diferenciales homog´ homog´eneas eneas de orden superior 101 3.9.3. Ecuaciones diferenciales no-homog´ eneas eneas de orden superior . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.10. 3.1 0. Apli Aplica caci cion ones es de las las ecu ecuac acio ione ness dife difere renc ncia iale less . . . . . . . . . . 110 110 3.10 3.10.1 .1.. Cir Circuit cuitos os el´ el´ectr e ctric icos os . . . . . . . . . . . . . . . . . . . 110 110 3.10.2. Movimiento libre no amortiguado . . . . . . . . . . . . 114 Ejem Ejempl plos os de mo movi vimi mien ento toss amo amort rtig igua uado doss . . . . . . . . . . . . 125 125 Movimiento forzado . . . . . . . . . . . . . . . . . . . . . . . 129 3.11. 3.1 1. Ecua Ecuaci cion ones es dife difere renc ncia iale less de de orde orden n supe superi rior or . . . . . . . . . . . 132 132 3.11 3.11.1 .1.. Con Con coefi coefici cieentes tes varia ariabl blees . . . . . . . . . . . . . . . . 132 132 3.11.2. Ecuacione Ecuacioness de segundo segundo orden no-homog´ no-homog´ eneas eneas . . . . . 134 3.12 3.12.. Sist Sistem emas as de Ecua Ecuaci cion ones es Dife Difere renc ncia iale less . . . . . . . . . . . . . . 136 136 3.13. Soluci´ on en en se serie de pot potencias . . . . . . . . . . . . . . . . . . 141
A. Sucesiones y series A.1. Su Sucesiones . . . . . . . . . . . . . . . . . . . A.2. Se Series . . . . . . . . . . . . . . . . . . . . . . A.2.1 A.2.1.. Propi Propied edad ades es de las seri series es conv conver erge gent ntes es A.3. Series geom´etricas . . . . . . . . . . . . . . . A.3.1 A.3.1.. Con Converg vergen enci cia a de una una seri seriee geo geom´ m´ etri e trica ca
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1 47 14 1 47 15 1 50 152 152 153 153
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Cap´ıtulo 1
INTEGRALES IMPROPIAS Hasta el momento se han estudiado integrales definidas, en donde el intervalo de integraci´on on es de longitud finita, o en donde la funci´on o n que se est´a integrando es acotada en dicho intervalo. El prop´osito osito es generalizar estas ideas ampliando el intervalo de integraci´on on a intervalos de longitud infinita, es decir intervalos en que uno de los extremos, o los dos, sean infinitos; o estudiar integrales sobre intervalos finitos pero en donde la funci´on no sea acotada ya sea en uno de los extremos del intervalo o en alg´un punto interior del intervalo de integraci´on. on. A este tipo de integrales se les llama integrales impropias.
1.1. 1.1.
Integ Integral rales es impr impropi opias as de de prime primera ra clas clase e
Consideremos en primer lugar la integral:
∞ − e
0
x dx. dx.
Cuya gr´afica, afica, es:
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2
CAP´ ITULO 1. INTEGRALES INTEGRALES IMPROPIA IMPROPIAS S
En este caso, como la funci´on on a integrar es positiva en el intervalo [ 0, 0 , ), podemos considerar que la integral representa el ´area area bajo la curva f ( f (x) = e−x . Para calcular dicha integral se calcula la integral definida 0b e−x dx en dx en donde b es cualquier n´ umero umero mayor que cero, b > 0 y luego luego se calc calcul ula a el l´ımit ım itee cuan cu ando do b b tiende tien de a infinito as´ as´ı:
∞
∞ −
b
x
e
dx = dx = l´ım b
→∞
0
− e
x
dx = dx = l´ım b
0
= 0 + 1 = 1 es decir:
→∞ ∞
1
− b x 0 e
= l´ım b
1
1 + 0 b e e
−
→∞
e−x dx = dx = 1. Se puede afirmar que:
0
∞
f ( f (x) dx, existe si la integral
c
b
f ( f (x) dx existe dx existe para todo b
c
b
l´ım
b
→∞
≥ c y
f ( f (x) dx existe dx existe y es finito. En este caso se escribe:
c
∞
b
→∞
f ( f (x)dx = dx = l´ım b
c
f ( f (x)dx.
c c
En forma similar se puede definir:
c
f ( f (x)dx = l´ım
−∞
a
→−∞
f ( f (x)dx.
a
En ambos casos se dice que la integral es convergente. En los casos en que los l´ımites no existan se dice que la integral es divergente. divergente.
Ejemplo 1.1.
∞ 2
1 dx = dx = l´ım b→∞ x(ln x)2
b
2
1 dx = dx = x(ln x)2 1 l´ım b→∞ ln x
− ∞ 2
1 1 1 = l´ım + = b→∞ ln b ln 2 ln 2
−
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1.1. INTEGRA INTEGRALES LES IMPROPIA IMPROPIAS S DE PRIMERA PRIMERA CLASE
3
Ahora, si los l os l´ımites ımites de integraci´ i ntegraci´on on son ambos infinitos se define la integral
∞ −∞∞
f ( f (x) dx, de la siguiente manera: c
f ( f (x) dx = dx =
−∞
f ( f (x) dx +
∞
f ( f (x) dx, c
c
−∞
∈R
siempre que las dos integrales de la derecha existan. En caso contrario se ∞ f (x) dx diverge. dice que la integral: −∞ f (
Ejemplo 1.3. f ( f (x) =
1 1 + x2
1
−2
∞
−∞
−1
1 dx = dx = 1 + x2
c
−∞
→−∞
c
a
2
Figura 2.
1 dx + 1 + x2
= l´ım a
1
∞ c
1 dx, c 1 + x2
1 dx + l´ım b→∞ 1 + x2
b
c
∈R
1 dx 1 + x2 b
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4
CAP´ ITULO 1. INTEGRALES INTEGRALES IMPROPIA IMPROPIAS S
1 f (x) = afica de la funci´ on f ( y halla hallarr el area ´ Ejemplo 1.4. Hacer la gr´ x bajo la curva en el primer cuadrante. f ( f (x) =
1 x 10 5
−4 −3 −2 −1 −5 −10 Figura 3. f (x) = Ejemplo 1.5. f ( .
x2
x +4
1
2
3
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1.1. INTEGRA INTEGRALES LES IMPROPIA IMPROPIAS S DE PRIMERA PRIMERA CLASE
5
Ejemplo 1.6. Hallar los valores de a y b, tales que:
∞ 1
2x2 + bx + a x(2x (2x + a)
−
1 dx = dx = 1.
Como el grado del polinomio del numerador es igual al grado del polinomio del denominador se efect´ua ua la divisi´on on y resulta:
∞ ∞
2x2 + bx + a (2x + a) x(2x
1
=
1
∞ − − − ∞ ∞ − − →∞ −− →∞ 1 dx =
1
(b a)x + a dx = dx = x(2x (2x + a)
−
1+
= l´ım c
= l´ım c
1
(b ( b a)x + a 2x2 + ax
A B + dx = dx = x 2x + a
1 ln x + (b 2
1 dx
1
− −
c
a
2)ln(2x 2)ln(2x + a )
ln(x ln(x) + ln(2x ln(2x + a)
1 (b 2
a 2)
c
1
1
= l´ım ln x(2x (2x + a) c
→∞
este l´ımite existe si y s´olo olo si 21 (b luego b luego b = a = a,, entonces,
1 b a 2 + dx x 2x + a
1 (b 2
−− a 2)
c
1
− a − 2) = −1, es decir, b − a − 2 = −2,
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6
CAP´ ITULO 1. INTEGRALES INTEGRALES IMPROPIA IMPROPIAS S
Ejercicios 1.7. Evaluar Evaluar cada una de las siguiente siguientess integrale integraless impropias: impropias:
1)
∞ − ∞ ∞ √ − −∞∞ −∞∞ − xe
2
x
dx
2)
4
3) 5)
0
1
7) 9) 11)
dx 2 (1 x) 3 1 dx ex 1 dx (5x (5x + 3)3 1 dx x2 + 8x 8x + 25
e
2x
x2 dx (x3 + 1)2 1 dx x ln x
0
1
2
∞ ∞ − −∞∞ √ −∞∞ −∞∞ | −|
sin(3x sin(3x)dx
4)
2
1
6) 8) 10) 12)
0
13) Para cierto valor de α, α , la integral
1 dx x3 1
x2 + 4 x dx ex
dx
t2 e t dt
0
∞ √ 0
1 1+2x 1+2x2
halle este, o estos valores y calc´ule ule la integral.
−
α x+1
dx es dx es convergente. 2
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7
1.2. INTEGRA INTEGRALES LES IMPROPIA IMPROPIAS S DE SEGUNDA SEGUNDA CLASE
1.2. 1.2.
Integ Integral rales es impr impropi opias as de de segun segunda da clas clase e
1.2. 1.2.1. 1.
Caso Caso 1:
En algunos casos se presentan integrales definidas en un intervalo semiabierto de la forma [ a, b ) con la propiedad propiedad siguiente: siguiente: l´ım f ( f (x) = . Si x
→b−
t
suponemos que la integral: b
t
f ( f (x)dx= dx= l´ım t
b
que
| ∞
≤ t < b,b , entonces
f ( f (x)dx, dx, siempre siempre que el l´ımite exista. En tal caso decimos decimos
→b− a
a
f ( f (x)dx existe dx existe para todo t todo t,, a
a
podemos definir, definir,
|
f ( f (x)dx, dx, es convergente. convergente. Si el l´ımite ımite no existe decimos que la integral es
a
divergente. En forma similar, se define la integral de una funci´on f on f ,, definida en un intervalo de la forma ( a, b ], con la condici´on on siguiente: siguie nte: l´ım f ( f (x) = . x
∞
+
→a
Ejemplo 1.8. 1
ln x dx = l´ım x →0+
√ 0
= l´ım
+
→0
ln x dx, si u = u = x
√ √ 1 x ln x − 4 x
√ 4
1
√
√ x
= l´ım
→0
+
− √ − √ 4 1
√
4 ln + 4 =
−4
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8
CAP´ ITULO 1. INTEGRALES INTEGRALES IMPROPIA IMPROPIAS S
Ejercicios 1.10. Calcular, si existe, cada una de las siguientes integrales impropias: 2
1)
(x
1
1
3)
1)
1 3
1
1
0
7
1
− √ − √ − √ −
x2
dx
2)
3
3
dx
4)
ln xdx
1
6)
0
1
7)
0
4
9)
0
1
11)
0
ln x dx x
x dx 16 x2 1 x dx x
2
4
8)
0
1
10)
1
2
12)
0
x
9
0
1
5)
1 dx x 3
√ − √ − − √ − √ − − √ −
x2
1
(x + 1)
4 3
dx
dx
1 dx 4 x
1 x dx 1+x 2 x dx x
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1.2. INTEGRA INTEGRALES LES IMPROPIA IMPROPIAS S DE SEGUNDA SEGUNDA CLASE 1 f (x)= √ Ejemplo 1.11. Sea f ( 3 2 x
27
−1
l´ım
0
1
√ 2 dx = dx = 3
x
1
√
→0− −1
3
27
1
√ −1
3
x2
dx +
3
0
27
√ 1 2 dx = dx = x
1
√ 2 dx = l´→ım0 δ →0 x2 x δ = l´ım 3 − (−1) + l´ım 3 →0 δ →0 dx + l´ım
+
−
1 3
3
−
1 3
+
−− → 1
3x 3 1
27 3
1
+ l´ım δ
1
0+
1
3x 3
27 δ
(δ ) 3 = 3 + 9 = 12
9
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10
CAP´ ITULO 1. INTEGRALES INTEGRALES IMPROPIA IMPROPIAS S
2
5)
0
1 3x
−4
2
6)
dx
1 2
7 ) Demuestre que l´ım h
+
−
h1
→ 0 −1
x
1
dx +
1 dx x 5 ln x
√ h
1 dx = 0. 0. x
En algunos casos es imposible calcular (si existe) el valor de la integral impropia directamente. Sin embargo se puede estudiar la convergencia o divergencia de ´esta esta mediante el uso de los m´etodos etodos indirectos conocidos, entre ellos los criterios de comparaci´on on directo e indirecto.
1.3. 1.3.
Criter Criterios ios de conv convergenc ergencia ia
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1.3. CRITERIO CRITERIOS S DE CONVERG CONVERGENCIA ENCIA
todo x ≥ 1, entonces ≤ 1 + x4 , para todo x x4 ≤ (1 + x4 )x4 , para todo x ≥ 1. 1 1 Luego 4 4 , para todo x todo x ≥ 1. ≤ x (x + 1) x4 ∞ 1 1 b 1 − Ahora, dx = l´ım = , es decir que:
En efecto, como 1
1
∞ ∞ 1
1
x4
→∞
b
3x3
1
3
1 dx es convergente y por el teorema 1.14, x4 1 dx, es convergente. 4 4 x (x + 1)
11
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12
1.4. 1.4.
CAP´ ITULO 1. INTEGRALES INTEGRALES IMPROPIA IMPROPIAS S
Funci unci´ on o ´n Gama
Una de las funciones usadas en teor´ teor´ıa de comunicaciones es la funci´on on gama que se denota con el s´ımbolo: ımbolo: Γ y se define de la siguiente manera: Γ(x Γ(x) =
∞
tx−1 e−t dt, para x para x > 0. 0 .
0
Mediante la integraci´on on por partes se concluye que: Γ(x Γ( x + 1) = x = x Γ(x Γ(x). En efecto, Γ(x Γ(x + 1) = du = du = x x tx 1
−
∞
tx e−t dt, dt, si hacemos u = tx y dv = e−t dt, dt, se tiene que,
0
dt y v = b
−e−t,
es dec decir ir que que,,
tx b
b
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´ GAMA 1.4. 1.4. FUNCI FUNCION
13
entera de a de a), ), entonces, usando la regla de L’hˆopital opital (n+1)veces (n+1)veces se tiene: ta a(a l´ım t = l´ım t→∞ e t→∞ a(a = l´ım t
→∞
− 1)(a 1)(a − 2).. 2)..((a − n)[a )[a − (n + 1)]t 1)]ta−(n+1) et 1)(a − 2).. 2)..((a − n)(a )(a − n − 1) − 1)(a =0 t(n+1)−a et
En particular, particular, tomando a = x = x + 1, es decir: deci r: l´ım Luego podemos escribir: 0 < 0 < Por lo tanto la integral
t
tx+1 et
1, para todo t
∞ − − ≤ ≤ ∞ tx
→∞
tx+1 et
1 e t dt
M
M
1 dt t2
= 0; 0;
M . ≥ M .
Como la integral de la derecha de la desigualdad es convergente, usando el criterio de comparaci´on, on, se concluye que la integral
∞
∞ − − tx
M 1
t
1 e t dt es
conver-
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14
CAP´ ITULO 1. INTEGRALES INTEGRALES IMPROPIA IMPROPIAS S
on directo o el criterio de Ejercicios 1.19. Usar el criterio de comparaci´ comparaci´ on por paso al l´ımite para para estudiar la convergencia convergencia de cada una de las siguientes integrales impropias
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16
CAP´ ITULO 2. TRANSFORM TRANSFORMADA ADA DE LAPLACE LAPLACE
= l´ım b
→∞
−
1
sesb
1 1 + 0 = , si s s > 0. 0 . se s
f ( f (t) = 1 F ( F ( )
1
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´ 2.1. INTRODUC INTRODUCCI CION
17
Se observa que la gr´afica afica se va trasladando hacia la derecha o izquierda seg´un un el valor de a de a..
NOTA
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18
CAP´ ITULO 2. TRANSFORM TRANSFORMADA ADA DE LAPLACE LAPLACE
=
Con Con esto esto se prue prueba ba que que la inte integr gral al
∞
(s
−
M , si s > c. c)e(s−c)T
f ( f (t)e−st dt exis existe te y por tant tantoo que que
0
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54
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
dy 2 = = 2 es deci decirr dx 1 que el ´angulo angulo de inclinaci´on on de la recta tangente tiene un ´angulo menor de 3π π 1350 pero obviamen obviamente te mayor mayor que 900 . 4 2 1350 con el eje horizo horizont ntal. al. En el punto punto ( 1, 2), 2),
−
−
−
Continuando de esta forma en cada punto de coordenadas enteras (por ejemplo) se puede construir el siguiente campo de tangentes y ’ = y/x 2
1.5
1
0.5
y
0
−0.5
−1
−1.5
−2 −2
−1.5
−1
−0. 5
0 x
0. 5
1
1. 5
2
Figura 68. Con un poco de esfuerzo se logra visualizar que las soluciones son rectas que pasan por el origen del sistema de coordenadas. dy = x 2 y, [ 2, 2] [ 2, 2] Ejemplo 3.2. dx dy En el el pun punto (0, (0, 1) se tie tiene ne que que = 0 es deci decirr la recta recta tange tangen nte a la dx dy soluci´on on F ( F (x, y ) = c es horiz horizon ontal. tal. En En el punto punto (1, (1, 2), 2), = 2 es dec decir ir dx la recta tangente tiene mayor inclinaci´on on que la recta y = x. = x.
−
×−
dy En el el pun punto to ( 1, 2), 2), = ( 1)2 2 = 2, es decir que la inclinaci´ inclinaci´on on de dx la recta tangente al punto inmediatamente anterior. Continuando con este proceso se logra construir el siguiente campo de tangentes:
−
− ×
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55
3.2. ESTUDIO CUALIT CUALITA ATIVO 2
y’=x
y
2
1.5
1
0.5
0
y
−0.5
−1
−1.5
−2 −2
−1.5
−1
−0. 5
0 x
0. 5
1
1. 5
2
Figura 69. Las funciones que son soluci´on on de esta ecuaci´on on diferencial son funciones exponenciales crecientes, estas se podr´an visualizar de una mejor forma si se construye el campo de tangentes sobre un rect´angulo angulo mucho mayor, como por ejemplo en el rect´angu a ngulo lo [ 10 10,, 10] [ 10 10,, 10]
−
×−
x’=xt
4
3.5
3
2.5
x
2
1.5
1
0.5
0
−2
−1.5
−1
−0. 5
0
0. 5
t
Figura 70.
1
1.5
2
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56
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
La anterior gr´afica afica es un ejemplo del campo de pendientes de la ecuaci´on diferencial x diferencial x = x.t, = x.t, generada generada por Matlab (igual que las dos gr´ aficas aficas anteriores a esta, con la instrucci´on: on: dfield6, luego enter y en la ventana que aparece se escogen los intervalos para las variables y se da la orden de procesar) en el cual t [ 2, 2] mient mientras ras que x [0, [0, 4]:
∈−
∈
Ejercici Ejercicios os 3.3. Construye el campo de pendientes para cada una de las siguientes ecuaciones diferenciales
1)
dy = x 2 + y dx dy x 2) = dx y
dy = xy dx dy 4) = x y dx
dy = xy dx dy 6) = xy 2 dx
3)
√
5)
√
dy on diferencial onoma = f ( f (x, y) se llama aut´ Definici´ on on 3.4. Una 3.4. Una ecuaci´ dx si f ( f (x, y) no depende de x, es decir si f ( f (x, y) f ( f (y)
≡
Ejemplo 3.5.
1)
dy =1 dx
− y2
2)
dy = (1 dx
− y)3
3)
dy = ln x dx
Estos modelos mo delos de ecuaciones aparecen con alguna frecuencia en la f´ısica, ısica, la definici´ on on que all´ all´ı se les da es la que una ecuaci´ ecuaci´on on diferencial es aut´onoma onoma si no depende de el tiempo. Son ejemplos de ecuaciones aut´onomas la ley del enfriamiento de Newton, el crecimiento de una poblaci´on, on, la mezcla de soluciones salinas, como se ver´a mas adelante. dy = f ( f (x, y ) es una ecuac ecuaci´ i´ on aut´ onoma, es decir Definici´ on on 3.6. Si dx si f ( entoncess decimos que un numero numero real real “c ”es un punto f (x, y) f ( f (y ), entonce critico de dicha ecuaci´ on y f ( cr´ıticos tambi´en en se les f (c) = 0. A los puntos cr´ llama puntos puntos de equilibrio o puntos estacionarios.
≡
dy = dx f ( f (y) enton entonces ces y = c es una soluci´on on constante de la ecuaci´on on aut´ onoma. onoma. Esta soluci´ soluci´ on se conoce como soluci´on on on de equilibro.
Observemos que si c es un cero de f ( f (y) en la la ecua ecuaci ci´´on on aut´ onoma onoma
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57
3.2. ESTUDIO CUALIT CUALITA ATIVO
cr´ıticos de cada ecuaci´ on diferencial Ejemplo Ejemplo 3.7. 3.7. Encuentre los puntos cr´ estable 1.
2.
dy = (1 y)3 aqu´ı f ( f (y) = (1 dx 0 1 y = 0 y = 1
− ⇔ −
⇔
− y)3, luego
⇔ (1 − y)3 =
f ( f (y ) = 0
dy = 1 y4 dx f ( f (y ) = 1 y4 entonces f ( f (y) = 0 1 y4 = 0 (1 y 2 )(1 + y + y 2 ) = 0 (1 y )(1 + y + y)(1 )(1 + y + y 2 ) = 0 y = 1 y y = 1 son son los los pun puntos tos cr´ıtic ıt icos os..
− − ⇔ −
⇔ − ⇔
⇔ − −
dy En una ecuaci´on on diferencial aut´onoma onoma = f ( f (y) los puntos punto s cr´ cr´ıticos son de dx especial importancia, ya que estos nos permiten hacer un an´alisis de como puede ser el comportamiento de las soluciones generales y las particulares. dy As´ As´ı por p or ejemplo e jemplo en la l a ecuaci´ e cuaci´on on aut´ onoma onoma = 1 y 2 se tienen dos puntos dx cr´ıtic ıt icos os:: y = 1 y y = 1. Si ubicamos ubicamos estos estos dos puntos puntos en una recta recta num´erica erica obtenemos obt enemos tres tr es intervalos: ( , 1), 1), ( 1, 1 ) y ( 1, 1, ). Se sabe que cualquier cualquier soluci´ soluci´on on es decre decrecie cient ntee en ( , 1), 1), pues pues 1 y 2 es menor que cero cero en ese inter interv valo alo,, crecien creciente te en ( 1, 1) y decre decreci cien ente te en (1, (1, ). Esta situaci´on on la podemos graficar (retrato de fase) dibujando una flecha hacia arriba en el intervalo donde es creciente y una flecha hacia abajo en el intervalo donde es decreciente.
−
−
1
−1
−∞ − − −∞ − −
−
∞
Creciente: flecha hacia arriba Decrecien Decreciente: te: flecha flecha hacia aba jo
Figura 71.
∞
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58
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
3.2.1. 3.2.1.
Puntos Puntos Atract Atractores ores y Repul Repulsore soress
Hay tres clases de comportamiento de una soluci´on on de la ecuaci´ ecuaci´ on on diferencial dy = f ( f (y ) cerca cerca de un punto punto critico critico “c”, a saber: dx 1. Cuando Cuando las puntas puntas de las flech flechas as en ambos ambos lados de “c” “c” apuntan apuntan hacia c, es decir leyendo de abajo hacia arriba creciente-decreciente, como en el caso y = 1 del ejempl ejemplo o anterior anterior,, en este este caso el punto punto se llama llama asint´ oticamente oticamente estable y c se llama punto atractor. 2. Cuando Cuando las puntas puntas de las flech flechas as apunta apuntan n alej´ alej´ andose de “c”decrecienteandose creciente, en este caso se dice que c es un punto inestable y a “c”se le llama punto repulsor, como en el caso y = 1 del ejemplo ejemplo anterio anterior. r.
−
3. Cuand Cuandoo las las pun puntas de las flec flechas has apun apunta tan n una una alej alej´ andose a´ndose de “c” y la otra acerc´andose andose a “c” creciente-creciente o decreciente-decreciente. En este caso no es ni atractor ni repulsor, se dice que “c”es semiestable.
cr´ıticos, Ejercicios 3.8. En los siguientes ejercicios encuentre los puntos cr´ construya el retrato fase de la ecuaci´ on diferencial y clasifique los puntos cr´ cr´ıticos en atractores, a tractores, repulsores o semiestables. semiest ables. 1. 2. 3. 4. 5. 6. 7. 8.
dy = y 3 y 2 dx dy = 2 y 2 3y dx dy = (y 1)3 dx
−
−
−
dy dx dy dx dy dx dy dx
= y 2
− 3y + 2
= y 4
− 16
= 2y(y
1)(y + 3) − 1)(y
= 4y
− y3
dy = (y dx
− 3)2
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3.3. 3.3. ESTUD ESTUDIO IO ANAL ANAL´ ITICO
3.3.
59
Estudio Anal´ Anal´ıtico
Inicialmente se consideran ecuaciones diferenciales de primer orden aquellas en las que se puede despejar y , es decir, ecuaciones en las que se puede escribir y = f ( f (x, y); uno de los casos particulares de dicha ecuaci´on es el caso en que f que f ((x, y) no depende de y, y , es decir que: y = g = g((x)
(3.1)
en donde g donde g es una funci´on on conocida. Resolver la ecuaci´on on 3.1 equivale equivale a encontrar una primitiva G(x); adem´as, as, los casos que interesan ser´an an aquellos donde G donde G((x) es continu continuaa en alg´ un un intervalo intervalo I I .. Se sabe que la integral integral indefiniindefinida de g es una primitiva y que cada primitiva se puede obtener de una ya encontrada sum´andole andole una constante, por tanto, la ecuaci´on on diferencial 3.1 tiene una infinidad de soluciones as´ as´ı:
y =
g (x) dx + C
Ahora, si queremos una soluci´on on particular, es decir, si necesitamos una funci´on on “y “ y ”que pase por alg´un un punto particular, digamos (a, (a, b), se reemplaza la x la x por p or a a,, la y la y por b y se despeja C , C , como por ejemplo:
Ejemplo Ejemplo 3.9. Si y = 3x2 . f (0) = 3. ecuaci´ on?. Adem´ as f (0)
¿Cual ´ es la funci´ on f que satisface satisface dicha
Integrando Integrando respecto a x se obtiene: y obtiene: y = x = x 3 + C , reemplazando x reemplazando x por 0, y, y y, y por 3 se concluye que C = C = 3. Luego la funci´on on requerida es y = x = x 3 + 3. En algunos casos la constante C C no se presenta en la soluci´on o n en forma aditiva sino en forma multiplicativa debido a equivalencias que se pueden apli ap lica carr as´ı: ı:
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60
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
2) y 2) y = x −2 + 2x, 2x,
y(1) = 5
3) y 3) y = (2x (2x + 1)4 ,
dy = 3x 3 x2 + 4x 4x dx x 5) y 5) y = , x+1 4)
y (0) = 6
− 1,
y (2) = 5 y (1) = 2
II) Comprobar que las ecuaciones diferenciales tienen las soluciones dadas:
y = 5e2x + 7e 7e−2x
1) y = 4y, 4 y,
y = 4e2x
2) xy 2) xyyy = (x ( x + 1)(y 1)(y + 1), 1) ,
dy y 3) = ; dx x 1 4) y 4) y y = 1; x 5) y 5) y + y 12 12yy = 0;
− − dy 6) y 6) y + dx
7) x 7) x 2
8) y 8) y 9)
d2 y dx2
− 3e−2x
x(y + 1) = 2e 2 ey−x x(y + 1) = 5e 5 ey−x
√
y = ( x + C )2 , x > 0 y = x = x ln x, x > 0 > 0 = C 1 e3x + C 2 e−4x y = C
2
= 0;
dy − x dx + 2y 2 y = 0;
9y − 9y = 0; − y + 9y
y = ln ln (x + C 1 ) + C 2 y = x = x cosln x, x > 0 > 0
y = C = C 1 sin sin 3x + C 2 cos cos 3x + 4e 4 ex ln(sec
)
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61
3.4. VARIABLES SEPARABLES SEPARABLES 1 y = sin x 2
13) y 13) y + y = sin x;
14)
dy y x = ; dx y + x
−
dy y x 15) = + ; dx x y y 16) y + x cot x
ln(x2 + y2 ) + 2 tan−1
y x
−
3.4. 3.4.
10e−x − 12 cos x + 10e
x
dy = 0; dx
x cos
y = C x
2
= 2ln x + C
||
y = C x
Variabl ariables es separa separable bless
Una ecuaci´ on on diferencial y diferencial y = f ( f (x, y ) en la que f que f ((x, y) se puede escribir como el producto de una funci´on on que depende de x : Q(x) con otra funci´on on que depende de y de y : R : R((y ) es decir y decir y = Q( Q (x)R(y) se llama de variables separables en donde tanto R tanto R((y ) como Q como Q((x) son funciones funciones conocidas y adem´ as as cont´ co nt´ınuas ınu as en todos los puntos de su dominio. Ahora, si R( R (y ) no es id´enticamente enticamente cero se puede dividir por R( R (y) para obtener:
1 1 y = Q( Q (x). Si hacemos = A( A (y), se obtiene obtiene : R(y ) R(y)
A(y )y = Q( Q (x). Si suponemos que y = Y = Y ((x) entonces A entonces A((Y ) Y )Y (x) = Q( Q (x).
Integrando Integrando respecto a x se obtiene:
A(Y ) Y )Y (x)dx = dx =
Q(x)dx + C, pero
Y (x)dx = dx = dy, dy,
entonces,
A(Y ) Y )dy = dy =
Ejemplo 3.12. 3
2
2
3
2
3
Q(x)dx + C
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62
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
ye−2y y = e x sin x, ⇒
ye −2y dy = dy =
ye −2y y dx = dx =
ex sin x dx + C,
⇒
ex sin x dx + C, luego luego
−y e−2y − e −2y = ex 2
4
2
sin x
− cos x
+ C
Ejemplo 3.14.
− ydx) ydx), ⇒ xdx + ydy = ydy = x x 2 ydy − xy 2 dx, ⇒ xdx + xy 2 dx = dx = x x 2 ydy − ydy, ⇒ (x + xy 2 )dx = dx = (x2 y − y)dy, como dy = dy = y ydx dx ⇒ (x + xy 2 )dx = dx = (x2 y − y)y dx ⇒ (1 + y 2 )x dx = dx = (x2 − 1)yy 1)yy dx ⇒ x y x y dx = dx = y dx, integrando integrando : : dx = dx = y dx ⇒ 2 2 2 x −1 1+y x −1 1 + y2 xdx + ydy = ydy = xy xy((xdy
x
x2
−1
dx = dx =
y 1 dy, luego ln(x ln(x2 1) + ln C 1 = 2 1+y 2 1 ln(1 + y2 ), es decir 1 + y 2 = C ( C (x2 2
−
− 1)
etodo etodo de separaci´ separaci´ on de variab variable less las Ejercici Ejercicios os 3.15. 3.15. Resuelva por el m´ siguientes ecuaciones diferenciales:
1) y 1) y = x 4 y−3
3)
− 1
x2 y + 1 + y 2 = 0
5) (x + 1)y 1)y + y 2 = 0
7) (x − 1) y = xy
−y tan y 4) xy (1 + x2 ) y − (1 + y 2 ) = 0 2) tan x cos y =
6) xyy x yy = 1 + x2 + y 2 + x2 y 2 dy 8) (x + 1) = x
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´ 3.5. 3.5. HOMOG HOMOGENEAS
63
Resuelva las ecuaciones diferenciales dadas sujetas a las condiciones iniciales que se indican: 15) sin x(e−y+1 ) dx = dx = (1 + cos x) dy , dy 16) + ty = ty = y, y, dt dx 17) = 4( 4 (x2 + 1), 1), dy
3.5.
y (0) = 0 y (1) = 3 π x( ) = 1 4
Homog´ eneas eneas
Se dice que una funci´on on f ( f (x, y ) es homog´enea enea de grado r si f ( f (tx,ty) tx,ty ) = tr f ( f (x, y) para alg´ un u n n´ umero umero real r real r..
Ejemplo 3.16. f ( f (x, y ) = x 2 + 3xy 3xy
− y2, ⇒
f ( f (tx,ty) tx,ty) = (tx ( tx))2 + 3(tx 3(tx)( )(ty ty))
ty )2 = t 2 x2 + 3t 3t2 xy − t2 y2 − (ty) = t 2 (x2 + 3xy 3xy − y 2 ) = t2 f ( f (x, y )
Ejemplo 3.17. f ( f (x, y ) =
x y x+y
− ⇒ f ( f (tx, tx, ty) ty) =
tx ty t(x y) x y = = = t 0 f ( f (x, y ) tx + ty t(x + y) x+y
−
−
−
Ejemplo 3.18. f ( f (x, y ) =
3
f ( f (tx, tx, ty) ty) =
x2 + y2 +
3
√ xy ⇒ 3
(tx) tx)2 + (ty (ty))2 +
3
(tx)( tx)(ty ty)) =
3
t2 x2 + t2 y2 +
3
(tx)( tx)(ty ty))
⇒
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64
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
on diferencial de la forma M(x,y)dx+N(x,y)dy=0 Teorema 3.19. Toda 3.19. Toda ecuaci´ donde M ( f uncioness homog´eneas eneas del mismo grado es una M (x, y ) y N ( N (x, y ) son funcione ecuaci´ on diferencial de variables separables. Prueba: Si en la ecuaci´on on diferencial de la forma M ( M (x, y)dx + dx + N N ((x, y)dy = 0 escribimos y = ux = ux , entonces dy = dy = udx udx + xdu , xdu , luego M ( M (x,ux) x,ux)dx + N ( N (x,ux)( x,ux)(udx udx + xdu) xdu) = 0 como las funciones M y N son homog´ hom og´eneas eneas de grado n tenemos xn M (1 M (1,, u)dx + xn N (1 N (1,, u)(udx )(udx + xdu) xdu) = 0, 0 , es decir M (1 M (1,, u)dx + N (1 N (1,, u)(udx )(udx + xdu) xdu) = 0, 0 , por lo tanto tanto:: [M (1 M (1,, u) + N (1 N (1,, u)u]dx + N (1 N (1,, u)xdu = xdu = 0, luego M (1 M (1,, u) + N (1 N (1,, u)u 1 = − du dx N (1 N (1,, u) x
on diferencial: (y 2 + yx) yx)dx + x2 dy = dy = 0. Ejemplo 3.20. Resolver la ecuaci´ Si escribimos y escribimos y = ux = ux,, 2 2
2
= udx + xdu, xdu, la ecuaci´on on dada se convierte en: ⇒ dy = udx 2
2
2
3
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´ 3.5. 3.5. HOMOG HOMOGENEAS
65
uxdx + x[ln x
− ln ux − 1][udx 1][udx + xdu] xdu] = 0, 0, ⇒ uxdx + x[− ln u − 1][udx 1][udx + xdu] xdu] = 0, 0, ⇒ 1]x du = du = 0 − u ln u dx − [ln u + 1]x ⇒ dxx + (ln uu +ln 1)u du = 0 ⇒ ln x + ln u + ln[ln u] + ln c = 0 ⇒ ln[cxu ln[cxu ln u] = 0 ⇒ cxu ln u = 1, reemplazando u cxy y ln = 1, 1 , como y(e)=1 ⇒ x x
c =
−1
Ejercicios 3.22. Resuelva cada una de las ecuaciones diferenciales dadas usando sustituciones adecuadas. 1) (x + y )dx + ydy = 0 dy y x 3) = dx y + x
−
y
5) [x2 e− x + y 2 ]dx = dx = xydy xydy
7)
dy x2 + 2y 2y 2 = dx xy
9) x 9) x 2 y + xy + xy + 2y 2y2 = 0
2) (y 2 + x2 )dx + x2 dy = dy = 0 dy x + 3y 3y 4) = dx 3x + y 6) (x2 + xy + xy + 3y 3 y 2 )dx
− (x2 + 2xy 2xy))dy = dy = 0
8) x 8) x y = y = y 10) y =
y
−
x2 + y 2
+ x sin
y
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66
3.6. 3.6.
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
Ecuaci Ecuacione oness dife diferen rencia ciales les exacta exactass
Del c´alculo alculo multivariado se sabe que si z = f ( f (x, y ) es una una fun funci ci´´on on de dos variables diferenciable, la diferencial de la variable dependiente dz llamada tambi´en en diferenci di ferencial al total t otal se define defi ne como: co mo:
dz = dz = d df f =
∂f ∂ f (x, y ) dx + (x, y ) dy ∂x ∂y
f (x, y ) = x 2 y Ejemplo 3.23. Si f (
− xy3 + xy entonces, df = (2xy (2 xy − y 3 + y) dx + (x (x2 − 3xy 2 + x) dy
M (x, y ) dx + dx + N N ((x, y) dy es on diferencial M ( Definici´ on on 3.24. Una expresi´ una diferencial exacta en una region R del plano xy si existe una una funci´ on = f ((x, y) de tal forma que df = M ( z = f M (x, y) dx + N ( N (x, y ) dy. Una ecuaci´on on diferencial de primer orden M ( M (x, y ) dx + dx + N N ((x, y ) dy dy = 0 es una ecuaci´on on diferencial exacta, si el lado izquierdo de la ecuaci´on o n es una diferencial exacta.
Ejemplo 3.25.
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3.6. ECUACIONES ECUACIONES DIFERENCIALES DIFERENCIALES EXACT EXACTAS AS
67
existen y son continuas en un conjunto abierto S entonces entonces se cumple que ∂ 2 f ∂ 2 f = ∂x ∂y ∂y ∂x Luego podemos afirmar que una ecuaci´on on diferencial de primer orden de la forma M ( M (x, y)dx + N ( N (x, y)dy = dy = 0 es una una ecuaci ecuaci´´on on exacta si M (x, y ), N ( N (x, y) dos funcione funcioness de dos variabl variables es Teorema 3.27. Sean M ( continuas y con derivadas parciales continuas en un conjunto abierto S. EnM (x, y)dx+ dx+N ( N (x, y )dy tonces, una condici´ on necesaria y suficiente para que M ( sea una diferencial exacta es que ∂M ∂N = ∂y ∂x
3.6.1.
M´ etodo etodo de soluci´ soluci´ on on de una ecuaci´on on exacta
Dada una ecuaci´on on diferencial de primer orden M ( M (x, y )dx + N ( N (x, y)dy = dy = 0 ∂M ∂N se determina si se cumple la ecuaci´on on = (siempre que las derivadas ∂y ∂x parciales parcia les tengan teng an sentido) sentido ) en caso afirmati a firmativo vo se tendr´ıa ıa que existe exi ste una funci´ f unci´on on ∂f f ( f (x, y) = c de tal forma que = M ( M (x, y ), si integramos, respecto a x ∂x se obtiene
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68
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
luego integrando la ultima ecuaci´on on respecto a y se encuentra encuentra la funci´ on on g (y) y finalmen finalmente te la soluci soluci´´on on impl´ impl´ıcita de la ecuaci´on on diferencial exacta es: f ( f (x, y) = c Es de anotar que la soluci´on on tambi´ tambi´ en en se encuentr encuentraa si se parte del hecho hecho ∂f que = N ( N (x, y) y se integ integra ra respec respecto to a “ y ”, para continuar con pasos ∂y similares a los que se han propuesto.
on diferencial Ejemplo 3.28. Resuelva la ecuaci´ (2x (2x + y ) dx + (x (x + 6y 6y ) dy = dy = 0 Aqu´ı M ( M (x, y ) = 2x 2x + y entonces, Si,
∂M =1 ∂y
y
y N ( N (x, y) = x + 6y 6y
∂N ∂M ∂N = 1, 1 , luego = ∂x ∂y ∂x
∂f = 2x 2 x + y, entonces ∂x
∂f dx = dx = ∂x
(2x (2x + y) dx + g(y )
f ( f (x, y ) = x 2 + xy + xy + g(y )
derivando respecto a y se obtiene
∂f = x + g (y ) ∂y
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3.6. ECUACIONES ECUACIONES DIFERENCIALES DIFERENCIALES EXACT EXACTAS AS
69
on diferencial Ejemplo 3.29. Resuelva la ecuaci´ (3x (3x2 y + ey ) dx + (x (x3 + xey Aqu´ı M ( M (x, y ) = 3x 3 x2 y + ey entonces,
∂M = 3x 3 x2 + ey ∂y
y
dy = 0 − 2y) dy =
y N ( N (x, y) = x 3 + xey
− 2y
∂N ∂M ∂N = 3x 3 x2 + ey , luego = ∂x ∂y ∂x
Esto significa que la ecuaci´on on original es exacta, luego existe una funci´on on f ( f (x, y ) = c
tal que
integrand integrandoo respecto respecto a y
∂f = N ( N (x, y ), es decir ∂y se obtiene obtiene
∂f = x 3 + xey ∂y
− 2y
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70
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
∂f luego existe una funci´on on f ( f (x, y ) tal tal que que = e x + y ; integrando integrando respecto respecto ∂x a x se obtiene f ( f (x, y) = e x + xy + xy + g(y ) derivando respecto a y se obtiene ∂f = x + g (y), luego ∂y 2 + x + ye y = x + g (y) es deci decirr g (y) = 2 + ye y , integrando integrando respecto a y se obtiene
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3.6. ECUACIONES ECUACIONES DIFERENCIALES DIFERENCIALES EXACT EXACTAS AS
71
Ejercicios 3.31. Determine si cada una de las siguientes ecuaciones diferenciales es exacta, en caso afirmativo a firmativo resu´ resu´elvala. elvala. (2xy + + 2) dy = dy = 0 1. y 2 dx + (2xy 2
2
2x2 y ey dy = dy = 0 2. 2x ey dx + 2x sin(xy))dx + x sin(xy sin(xy)) dy = dy = 0 3. y sin(xy (2x + y ) dx + (x (x 4. (2x (2xy 2 5. (2xy
6.
dy = 0 − 2y) dy =
(2x2 y + 2) dy = dy = 0 − 2) dx + (2x
2x dx 2y dy + =0 (x2 + y 2 )2 (x2 + y 2 )2
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72
3.7. 3.7.
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
Line Lineal ales es de prim primer er orde orden n
Una ecuaci´ ecuaci´ on diferencial lineal de primer orden es de la forma: on y + P ( P (x)y = Q = Q((x) donde P ( P (x) y Q(x) son funcione funcioness definidas definidas y continu continuas as en alg´ alg´ un un intervalo I. Un caso particular es cuando Q(x) = 0 cuy cuya sol soluc uci´ i´ on on se encuentra por separaci´ on de variables. Ahora, si Q(x) = 0 nos damos on damos cuent cuenta a que la parte parte izquierda de la ecuaci´on on en menci´ on es casi la derivada de un producto. Si on P (x)dx obtenemos: multiplicamos la ecuaci´on on por e P (
P ( P (x)dx
ye
+ P ( P (x)ye
P ( P (x)dx
P ( P (x)dx
P ( P (x)dx
= Q( Q (x)e Q( Q ( )
, que podemos escribir:
P ( P (x)dx
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73
3.7. LINEALES LINEALES DE DE PRIMER PRIMER ORDEN ORDEN on diferencial: Ejemplo 3.33. Resolver la ecuaci´ 2
x(x + 1)y 1)y + y = x = x((x + 1)2 e−x dividiendo por x(x + 1)
se obtiene: obtiene:
1 x(x + 1)2 −x2 y = y = e luego, x(x + 1) x(x + 1) 1 2 P ( P (x) = , Q(x) = (x ( x + 1)e 1)e−x , por lo tanto x(x + 1) 1 x P ( P (x) dx = dx = dx = dx = ln , x(x + 1) x+1 y +
x y = e = e − ln x
+1
x+1 y = x
⇒
x2 ln
(x + 1)e 1)e− e
2
x x+1
x(x + 1)e 1)e−x x+1 dx + C = x+1 x
dx + C
2
xe−x
dx + C , luego
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74
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
Ejercicios 3.35. Resolver cada una de las siguientes ecuaciones diferenciales: 1) y 1) y 3)
= e 2x − 3y = e
2
sin
2) y 2) y sin x + y cos x = 1 4)
2
5
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3.7. LINEALES LINEALES DE DE PRIMER PRIMER ORDEN ORDEN Aqu´ı, n = 1/2, y v = y = y 1/2 v 2v = e = e x , P ( P (x) = 2, Q(x) = e x , luego v = e = e 2x ex e−2x dx + C = e 2x e−x dx + C v = e = e 2x ( e−x + C ) = C e2x ex , como v como v = y = y 1/2 , x )2 (C 2x
−
−
−
−
⇒
75
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76
3.8. 3.8.
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
Prob Proble lema mass d de e apl aplic icac aci´ i´ on on
Ejemplo 3.39. En el estudio de crecimiento de una poblaci´on on (humana, animal o bacteri-
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´ 3.8. PROBLE PROBLEMAS MAS DE APLICA APLICACI CION di + 3i 3 i = 6, luego dt i(t) = e −3t 6e3t dt + C i( i (t) = 2 + C e−3t . Como i Como i(0) (0) = 0 i( i (t) = 2 2e−3t
⇒
Ejemplo 3.41.
−
77
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78
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
Ejemplo 3.43.
Ley de enfriamiento de Newton La variaci´ ariaci´on o n de la temperatura respecto al tiempo de un cuerpo que se
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114
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
3.10.2. 3.10.2.
Movimi Movimien ento to libre libre no no amortig amortiguado uado
Segunda ley de Newton Al colocar una masa m en un resort resortee que est´ est´a colgado colga do de un muelle ´esta esta lo estira una longitud longitud s y luego llega llega a una posici´ posici´on on de equilibrio, en la que su peso w encuentr encuentraa la estabilidad estabilidad por la fuerza fuerza de restituci´ restituci´ on on k s como se muestra en la figura siguiente
s
m
Posici´ on on de equilibrio mg ks = ks = 0
−
Figura 73.
Luego, si la masa m se desplaza una distancia x respecto respecto de su su posici´ posici´on on de equilibrio la fuerza de restituci´on on del resorte es k (s + x + x). ). Supon Suponie iend ndoo que no hay fuerzas de retardo o rozamiento que act´uen uen sobre el sistema y que la masa se mueve libremente de otras fuerzas externas, entonces podemos, utilizando la segunda ley de Newton establecer la siguiente ecuaci´on: on: m
d2 x = dt2
mg = −kx + mg − kx = kx = −kx, es decir −k(s + x) + mg = d2 x m 2 = dt
−kx
(3.17)
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
115
m m m
m m
m m
m
Figura 74. m Cuya gr´afica afica en el plano cartesiano, tiempo vs desplazamiento es
•
•
A
B
•
C
Figura 75.
A: Posici´ Posici´ on de equilibrio. Inicia el movimiento. on B: Posici´ Posici´ on de equilibrio (la masa m est´a de vuelta) en direcci´on on on arriba debido a la fuerza restablecedora C: Posici´ Posici´ on de equilibrio ( la masa m pasa por el origen on origen debido debido a la fuerza fuerza ejercida inicialmente ). Y el movimiento contin´ua ua “ indefinidamente”.
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116
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
Si en la ecuaci´on on (3.17), de la p´agina agina 114 se divide divi de cada ca da t´ermino ermino por m se obtiene d2 x k + x = 0, dt2 m Como k y m son positivos, p ositivos, podemos escribir ´esta esta ultima u ´ ltima ecuaci´on on d2 x k 2 2 + w x = 0 , donde w = dt2 m
(3.18)
Se dice que la ecuaci´on on (3.18) describe el movimiento arm´onico onico simple m.a.s o movimiento libre no amortiguado. Las condiciones iniciales asociadas a la ecuaci´on on diferencial son x(0) = x0 cantidad cantidad de desplazamie desplazamiento nto inicial inicial y x (0) = x1 la velocida velocidad d inicial inicial de la masa. masa. Por ejempl ejemploo si x0 > 0 y x1 < 0, la masa masa part partee de un punto punto abajo abajo de la posici posici´ on ´on de equilibrio con una velocidad x1 dirigida dirigida hacia arriba arriba.. Cuando x1 = 0, la masa m parte del reposo, por ejemplo, x0 < 0, x1 = 0, la mas masaa part partee del del repos reposoo desde un punto ubicado x0 , unidad unidades es arri arriba ba de de la posici´ posici´ on on de equilibrio. 2 2 Se sabe que la ecuaci´on on auxiliar m + w = 0, 0, tiene tiene como como soluci solucione oness a los los n´ umeros umeros complejos m1 = w i y m2 = wi , as´ as´ı la soluci´ on on general de la ecuaci´on on (3.18) es:
| |
−
x(t) = c 1 cos(wt cos(wt)) + c2 sin(wt sin(wt)) El per´ per´ıodo de las vibraciones libres que describe la funci´on on x(t) es la frecuencia 2wπ .
2π w
y
Ejemplo 3.74. Una masa que pesa 4lb. hace que un resorte estire 10 pulgadas. Cuando t=0, la masa se suelta desde un punto que est´ a a 10 pulgadas abajo de la posici´ on de equilibrio con una velocidad inicial, hacia arriba, de 4 Deduz duzcca la ecuaci´ cuaci´ on del movimiento libre 5 pies/seg . De Como se emplea el sistema t´ecnico ecnico de unidades inglesas, las medidas expresadas sadas en pulgada pulgadass se debe expresar expresar en pies pies sabien sabiendo do que 12pulg= 12pulg= 1pie, 1pie, entonces entonces 10pulg= 56 pies. Adem´as as se debe convertir las unidades de peso, que est´an an en libras, en unidades de masa, sabiendo que m = wg , en est estee 4 caso m = 32 = 81 slug. Tambi´en en seg´ seg u ´n la le ley de Hoo Hook k 4 = k 56 , k = 24 on (3.17) se convierte en: on 5 lib/pie. Luego la ecuaci´
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
1 d2 x = 8 dt2
192 , o sea w = 5 soluci´on on general
192 , 5
d2 x 192 1 92 + x = 0 5 dt2
− 245 x, o
Con las condiciones iniciales x(0) =
117
− 56
y x (0) =
− 45 ,
ento entonc nces es w2 =
de modo modo que que la ecuac ecuaci´ i´ on diferencial tiene como on
x(t) = c 1 cos
192 t 5
+ c2 sin
192 t 5
5 5 Como x(0) = , ento entonc nces es = c 1 . Ahora 6 6 x (t) =
−
Como x (0) =
192 c1 sin 5
− 45 ,
enton entonces ces
192 t 5
− 45 =
√ c2 = − √
+
192 c2 cos 5
192 t 5
192 c2 ; es deci decirr 5
4 5 = 5 192
− √ 160
Luego la ecuaci´on on del movimiento es: 5 x(t) = cos 6
192 t 5
1 sin 60
− √
192 t 5
(3.19)
Otra forma alternativa para x(t). Cuan Cuando do c 1 = 0 y c2 = 0 y sabi sabien endo do 2 2 que que sin( sin(α α + β ) = sin α cos β + + cos α sin β β podemos hacer A = c1 + c2 y φ un ´angulo angulo de fase definido as´ as´ı:
c1 A c2 cos φ = A sin φ =
tan φ =
c1 c2
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118
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
donde,
φ =
arctan arctan arctan arctan
c1 c2 c1 c2 c1 c2 c1 c2
,
si c si c 1 > 0 > 0 y c 2 > 0 > 0
+ π,
si c si c 1 > 0 > 0 y c 2 < 0 < 0
+ π,
si c si c 1 < 0 < 0 y c 2 < 0 < 0
+ 2π, 2π,
si c si c 1 < 0 < 0 y c 2 > 0 > 0
Entonces Entonces la ecuaci´ ecuaci´ on on
x(t) = c 1 cos(wt cos(wt)) + c2 sin(wt sin(wt)) con las anteriore anterioress consideraci consideraciones ones se transforma transforma en: x(t) = A sin φ cos(wt cos(wt)) + A cos φ sin(wt sin(wt)) x(t) = A[sin A [sin φ cos(wt cos(wt)) + cos φ sin(wt sin(wt)] )] x(t) = A sin(wt sin(wt + φ)
En la ecuaci´on on (3.19) tenemos A = A = A =
25 1 + = 36 60
8 √ 3 10
x(t) =
250 + 6 = 360
√ 5 6
2
+
1 60
2
;
256 16 = ; 360 6 10
√
y φ = 1,7244 rad. y as´ as´ı
8 √ sin(wt sin(wt + 1, 1,7244) 3 10
El modelo de movimiento movimiento libre arm´onico onico para una masa que se est´a moviendo en un resorte no es realista porque el movimiento que describe la ecuaci´on (3.17) supone que no hay fuerzas externas que act´uen uen en el sistema lo cual es cierto s´olo olo cuando el sistema est´e en el vac´ vac´ıo perfecto. perfe cto. Es por esto que el modelo (3.17) hay que refinarlo, suponiendo que las fuerzas de amortiguamiento que act´ uan sobre el cuerpo son directamente proporuan cionales a la velocidad instant´anea, anea, es decir, la ecuaci´on on (3.17) se puede
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
119
modi mo difica ficarr escr e scrib ibi´ i´endol end olaa as´ı: ı: d 2 x dx = kx β (3.20) dt2 dt donde β es β es una constante de amortiguamiento positiva y el signo negativo es consecuencia del hecho que la fuerza amortiguadora act´ua ua en direcci´on on opuesta a la del movimiento. La ecuaci´on on (3.20) se puede escribir m
m
− −
d 2 x dx + β + kx = kx = 0 dt2 dt
dividiendo dividiendo por m (m = 00)) se obt obtie iene ne
k como = w 2 m as´ı
d2 x β dx k + + x = 0 (3.21) dt2 m dt m β y si 2λ = entonces la ecuaci´on on (3.21) se puede escribir escribir m d2 x dx + 2λ 2λ + w2 x = 0 2 dt dt
(3.22)
ahora la ecuaci´ on on auxiliar de (3.22) es m2 + 2λ 2λ m + w 2 = 0 esta ultima u ´ltima ecuaci´on on tiene como soluciones
−2λ ± m =
(2λ (2λ)2 4w 2 2 4λ2 4w2 2 λ2 w 2
−
√ − − 2λ ± m = m =
−λ ± − √ m1 = −λ + λ2 − w2 y
− − √ −
las soluciones son m2 = λ λ2 w2 luego podemos distinguir tres casos en forma an´aloga aloga como en la ecuaci´on on (3.6).
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120
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
√ −
debido a que λ2 w 2 < λ2 entonces λ2 w2 < λ de lo que se puede puede deduci deducirr que λ + λ2 w2 < 0 y tambi´ ta mbi´en en λ λ2 w2 < 0 de lo que podemos concluir que (3.23) es una funci´ on on decreciente como
−
−
√ −
√ −
− −
x(t)
t
Figura 76. o puede tener un punto m´ m´ınimo y ser negativa en un intervalo intervalo de la form formaa [a, ). Su gr´afica afica puede ser:
∞
x(t)
t
Figura 77. caso
II: I I: λ2 w 2 = 0. En este caso caso se dice que el el sistema sistema est´a cr´ıtic ıt icam ament entee amortiguado puesto que cualquier disminuci´on on peque˜ na na de la fuerza de amortiguamiento originar´ originar´ıa un movimiento movimiento oscilatorio. La soluci´ on on general de la ecuaci´on on (3.22) es:
−
x(t) = c 1 em1 t + c2 t em1 t , es decir x(t) = e −m1 t [c1 + c2 t] cuyas gr´aficas aficas pueden ser:
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
x(t)
121
x(t)
t
Figura 78.
t
Figura 79.
x(t)
t
Figura 80.
El movimiento es oscilatorio pero a causa del factor e −λ t , las amplitudes de las vibraciones tienden a cero.
Ejemplo 3.75. Consid Considere eremos mos un reso resorte rte al al que un peso peso de 5 lb lo alarga alarga en en 21. pie. pie. El peso se sujeta al resorte y este alcanza el equilibrio, luego el peso es halado hacia abajo 3 pulgadas desde el punto de equilibrio, entonces vuelve hacia arriba a una vel veloci ocida dad d de 5 pis Encuentre tre una una ecuaci´ ecuaci´ on on que d´e la posici´on on del seg . Encuen peso en cada momento. 1 Se sabe que F kx k x ento entonc ncees 5 k k 10 lib se sabe sabe ade adem´ m´ as as que
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122
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
La ecuaci´on on auxiliar es m2 + 64 = 0 lueg luegoo m1 = 8i, m 2 =
−8i es decir que la soluci´on on es
x(t) = c 1 cos(8t cos(8t) + c2 sin(8t sin(8t) como como 1 1 x(0) = , = c 1 4 4 ahora x ahora x (t) = 8c1 sin(8t sin(8t) + 8c 8 c2 cos(8t cos(8t) 3 x (0) = 6, 6, 6 = 8c 8 c2 , c2 = 4
−
−
−
Y as´ as´ı la ecuaci´ ecua ci´on on que da la posici´on on del cuerpo en cada instante es: 1 x(t) = cos(8t cos(8t) 4
sin(8t) − 34 sin(8t
Ejemplo 3.76. Un resorte es alargado 3 pul por un peso de 10 lb, suponga que el peso es halado 4 pul abajo de la posici´on on de equilibrio. Se le imprime una velocidad inicial de 5 pis Encuentree la ecuaci´ on que describe el movimiento en cada on seg . Encuentr instante. 1 F = kx; kx ; 2 = k, 4
lb k = 8 pie
ahora mx (t) + λx( λx(t) = 0, 0 , entonces 2 x (t) + 8x 8 x(t) = 0 32 Es decir que la ecuaci´on on diferencial del sistema es: x (t) + 128x 128x(t) = 0, x(0) = la ecuaci´on on auxili auxiliar ar es
1 , 3
x (0) = 5
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
123
1 1 como x como x(0) (0) = , entonces c1 = , ahora 3 3 x (t) = c1 8 2 sin( sin(88 2 t) + c2 8 2 cos( cos(88 2 t) 5 y como x como x (0) = 5, 5, entonces c entonces c 2 = 8 2
− √
√
√
√
√
y la ecuaci´on on que describe el movimiento es:
√
√
1 5 x(t) = cos(8 2 t) + sin(8 2 t) 3 8 2
√
Ejemplo 3.77. A cierto cierto resor resorte te un peso de 8 lb lo alarg alargaa 6 pul. Supon Suponga ga que al resor resorte te se le suje sujeta ta un peso peso de 4lb 4lb y despu despu´´es e s se llev llevaa 2 pulg pulgad adas as arri arriba ba de la posici´on on de equilibrio, para luego soltarlo con una velocidad negativa de pul 15 seg . ¿Cu´ando ando llegar´ llegara´ a su punto m´as as bajo por primera vez? 1 lb Tenemos F = kx; kx ; 8 = k ; lueg luegoo k = 16 2 pie ahora mx ahora mx (t) + k x(t) = 0 enton entonces ces 4 x (t) + 16x 16x(t) = 0 enton entonces ces 32 x (t) + 128x 128x(t) = 0 , x(0) =
1 ; x (0) = 6
−15 pies seg
Luego la soluci´on on general de la ecuaci´on on es:
√
√
x(t) = c 1 cos(8 2t) + c2 sin(8 2t) 1 Como x(0) = ; y x (0) = 15 Enton Entonces ces 6 1 15 c1 = y c2 = 6 8 2
− −
− − √
Luego la ecuaci´on on que da la posici´on on en cada instante es:
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124
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
A =
2153 46 46,,4 = 1152 33 33,,9
A = 1,3670
Esto quiere quiere decir que el cuerpo cuerpo oscila 1.3670 pies hacia hacia arriba y 1.3670 pies hacia abajo. La ecuaci´ on on (3.24) se puede escribir as´ı: ı:
√
x(t) = 1, 1 ,367 36700 sin(8 sin(8 2 t + 3, 3,2666) El espacio recorrido hasta llegar al punto m´as as bajo:
−1,3670 −1 √ 8 2 t + 3, 3,2666 √
√
= 1,367 36700 sin(8 sin(8 2 t + 3, 3,2666)
√
= sin(8 2 t + 3, 3,2666 2666)) Enton Entonces ces = 4,712 71233 es decir decir
8 2 t = 1,6973 t = 0,127 seg
Luego el espacio recorrido para alcanzar la posici´on on de equilibrio desde el punto m´ as as alto es 0.1 0.127 27 seg. seg. Ahora, Ahora, el tiempo tiempo necesar necesario io para alcanz alcanzar ar la posici´on on m´as as alta desde el sitio donde fue lanzado es:
√
1,2003 = 1,3670 sin(8 sin(8 2t + 3, 3,2666) es decir
∼
t = 0,10 Luego el tiempo total para alcanzar el punto m´as a s bajo es 0.35 0.35 seg seg apro aproxi xi--
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
125
3. Encuen Encuentre tre la corr corrien iente te i(t) para para un circuito circuito L-R-C L-R-C en serie serie si L = 3,5 H, R = 1000Ω, 1000Ω, C = 2 10−6 F y E (t) = 120 sin(377 sin(377tt)v .
×
4. Un res resor orte te es ala alarg rgad ado o 8 pulgad pulgadas as por por un peso peso de 5 lb, lb, supon suponga ga que que el peso peso es llev llevado ado 6 pul abajo abajo de la posic posici´ i´ on de equilibrio, donde se on pie imprim imprimee una velocid velocidad ad hacia hacia abajo de 10 seg .
a ) Encuen Encuentre tre una ecuaci ecuaci´´on on que describa el movimiento. b ) ¿Cu´al al es la amplitud de dicho movimiento? c ) ¿Cual ´a l es su ´angulo angulo de desfase? 5. Un resor esorte te es ala alarg rgad adoo 1 pie pie por por un un peso peso de 15 lb, lb, supon suponga ga que que el peso peso es llev llevado ado 6 pul abajo abajo de la posic posici´ i´ on de equilibrio, donde se on pie imprim imprimee una veloci velocidad dad de 10 seg . Encuentre tre una ecuaci ecuaci´´on on que describa el movimiento. a ) Encuen al es la amplitud de dicho movimiento? b ) ¿Cu´al 6. A un resort resortee que es alarg alargado ado 6 pulgad pulgadas as por un peso de 5 lb, lb, se le pega una pesa de 10 lb, luego de alcanzar alcanzar la posici´ on de equilibrio, se lleva 8 on pulgadas arriba de la posici´on on de equilibr equilibrio io donde donde se le imprime imprime una
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126
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
fuerza de amortiguamiento de 41 de su velocidad. velocidad. Encuen Encuentre tre la ecuaci´ ecuaci´on on que describe la posici´on on del cuerpo cuerpo en el instan instante te t. F = kx; kx ; 4 = 0,64 k;
∗
k =
4 , 0,64
k = 6,25
Luego la ecuaci´on on diferencial que describe el movimiento es: 1 mx (t) + x (t) + 6, 6 ,25 25x x(t) = 0 4 4 1 x (t) + x (t) + 6, 6 ,25 25x x(t) = 0 32 4 x (t) + 2x 2 x (t) + 50x 50x(t) = 0 la ecuaci´on on auxiliar es m2 + 2m 2m + 50 = 0 y sus ra´ ra´ıces son
√ 2 ± 4 − 200 − m = ,
m1 =
2 1 + 7i 7 i y m2 =
−
−1 − 7i la soluci´on on es
x(t) = c1 e−t cos(7t cos(7t) + c2 e−t sin(7t sin(7t) como como 1 1 x(0) = , entonces c1 = ahora 3 3
−
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
127
Ejemplo 3.80. Un resorte resorte es alargado alargado 4 pulgadas por un peso de 2 lb. El peso baja desde el punto de equilibrio a una velocidad positiva de 12 pie seg . Si la resistencia del aire opone una fuerza fuerza de magnitud magnitud 0.02 la velocidad, velocidad, describ describa a el movimien movimiento. to. 1 F = kx; kx ; 2 = k ; k = 6, luego 3 2 m x(t) + x(t) + 6x 6 x(t) = 0 100 2 2 x(t) + x(t) + 6x 6 x(t) = 0 32 100 x(t) + 0, 0 ,32 32x x(t) + 96x 96x(t) = 0 La ecuaci´on on auxiliar es m2 + 0, 0,32 32m m + 96 = 0 Las soluciones de dicha ecuaci´on on son:
(0,32)2 − 4,96 −0,32 ± (0, m = 2 − −0,32 − 19 0,32 + 19, 19,59 59ii 19,,59 59ii m1 = y m2 =
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128
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
12 = 19, 19,59 c2 ; c2 =
12 = 0, 0 ,612 19 19,,59
de esta forma 32t x(t) = 0, 0 ,612 612ee−0,32t sin(19, sin(19,59 t)
Ejercicios 3.81. 1. Un resor resorte te es alarg alargado ado 0.4 0.4 pies por un un peso de 4 lb. El peso peso se sujet sujetaa al resorte y se permite que el sistema alcance el equilibrio, entonces se imprime imprime al peso una velocid velocidad ad hacia arriba arriba de 2 pie seg . Suponga que el movimiento se efect´ua u a en un medio que se opone una fuerza de magnitud igual a la velocidad del peso en movimiento. Determine la posici´on on del peso como funci´on on del tiempo. 2. Resuelva Resuelva el ejercicio anterior si si despu´ es de alcanzar el equilibrio el peso es se llev llevaa 6 pulgad pulgadas as arriba arriba de de este. este. 3. Resuelva Resuelva el primer ejercicio si despu´ despu´es es que el peso alcanza el equilibrio se lleva a una posici´on on de 6 pulgadas abajo de la posici´ on on de equilibrio y se imprime imprime una una velocid velocidad ad de 2 pies seg .
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
129
8. Resuelv Resuelva a el ejercicio ejercicio anterior anterior si si el cuerpo se lleva lleva 8 pulgadas arriba arriba del punto punto de equilibrio equilibrio y la velocidad que se le imprime imprime es de 7 pie seg hacia abajo. lb 9. Una Una mas masaa de de 11Kg Kg est´ est´ a unida a un resorte cuya constante es 16 pie y todo el sistema se sumerge en un l´ıquido ıquido que imparte una fuerza de amortiguaci´ on on num´ num´ericament ericamentee igual a 10 veces veces la velocidad velocidad instant´ instant´ anea. anea. Formule la ecuaci´on on del movimiento si:
a ) El contra contrapeso peso se suelt suelta a partien partiendo do del reposo reposo a 1 pie bajo la posici´ on on de equilibrio. b ) El contr contrape apeso so se suelt sueltaa 1 pie pie abajo de la posici posici´´on on de equilibrio pies con una una veloc velocida idad d de 12 seg hacia arriba.
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130
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
peso de 16 lb estira estira 8/3 pies pies un resorte. esorte. Al Al principio principio Ejemplo 3.82. Un peso en contrapeso parte del reposo a 2 pies abajo de la posici´ on de equilibrio y el movimiento ocurre en un medio que presenta una fuerza de amortiguamiento num´ num´ericament ericamentee igual a la mitad de la magnitud magnitud de la velocidad. velocidad. De Deduzc duzca a la ecuaci´ on del movimiento si el contrapeso est´ a impulsado por una fuerza f (t) = 10 cos( cos(33t). externa igual a f (
lb F = k x 16 = 8/ 8/3 k k = 6 pie adem´ as x as x(0) (0) = 2 y x y x (0) = 0. 0. ahora, como m x (t) + β x (t) + λ x(t) = 10 cos( cos(33t), entonces
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3.10. 3.10. APLICA APLICACION CIONES ES DE LAS ECUACION ECUACIONES ES DIFERENCIALES
131
aplicando la transformada inversa se tiene t/2 x(t) = 2 e−t/2 cos
10 sin(3 t) 3
√ 47 t 2
2 −t/2 + e t/2 sin 47
t/2 cos − 103 e−t/2
√ √ 47
√ 47 t 2
10 cos(3 t)+ 3
√
35 2 −t/2 47 √ − e t/2 cos t 2 3 47 2 √ 47 4 −t/2 x(t) = − e t/2 cos t − 3 2 √ 47 10 64 −t/2 √ e t/2 sin t + (cos(3 t) + sin(3t sin(3t)) 2 3 3 47
Ejercicios 3.83.
+
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132
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
3.11. 3.11.
Ecuaci Ecuacione oness difere diferenci nciale aless de orden orden superi superior or
3.11.1. 3.11.1.
Con coeficien coeficientes tes variables ariables
Definici´ on on Una ecuaci´ ecuacion o´n diferencial de orden superior y con coeficientes variables es de la forma: an (x)y (n) + an−1 (x)y (n−1) + . . . + a1 (x)y + a0 (x)y = f = f ((x) donde las funciones ai (x), i = 1, 2, 2 , . . . , n y f ( f (x) est´ stan ´an definidas y son continuas en un intervalo I . Un caso particular de dichas ecuaciones lo constituye las llamadas ecuaciones de Cauchy-Eu Cau chy-Euler ler aqu´ı
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3.11. 3.11. ECUACION ECUACIONES ES DIFERENCIA DIFERENCIALES LES DE ORDEN SUPERIOR
133
yh = c = c 1 xm1 + c2 xm2 b) que tenga una ra´ ra´ız m real de multiplicidad dos (tiene ra´ıces ıces iguales). Se m m puede probar que u1 = x y u2 = x = x ln x son dos soluciones linealmente independientes y que por tanto la soluci´on on general de (3.25) es: yh = c = c 1 xm + c2 xm ln x c) que tenga dos ra´ ra´ıces complejas la una conjugada conjugada de la otra, digamos: m1 = α + β i y m2 = α β i en este caso u1 = xα cos(β cos(β ln ln x) y u2 = xα sin(β sin(β ln ln x) son dos soluci solucione oness lineal linealmen mente te indepen independie dient ntes, es, luego luego se concluye que la soluci´on on general es:
−
y
c xα cos(β cos(β ln ln x) + c xα sin(β sin(β ln ln x)
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134
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
I I I ) x2 y + 5xy 5 xy + 4y 4 y = 0 Aqu´ı a = 1, b = 5,
c = 4 . La ecuaci´on on auxiliar es: m 2 + 4m 4 m + 4 = 0
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3.11. 3.11. ECUACION ECUACIONES ES DIFERENCIA DIFERENCIALES LES DE ORDEN SUPERIOR
135
variaci´on on de par´ametros ametros teniendo en cuenta que se debe escribir (3.27) de la forma y +
b c f ( f (x) y + 2 y = ax ax ax2
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136
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
ones diferenciales: Ejercicios 3.87. Resuelva las siguientes ecuaci´
1) x 1) x 2 y
6 y = ln x2 − 4xy + 6y
2) x 2) x 2 y
= x 2 ex − 6y = x
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3.12. 3.12. SISTEMAS SISTEMAS DE ECUACION ECUACIONES ES DIFERENCIAL DIFERENCIALES ES 137
{ dx } = L {x − y} dt
L
{ dy } = L {2x + y} dt
L
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138
CAP´ ITULO 3. ECUACIONES ECUACIONES DIFERENCIA DIFERENCIALES LES
− − − −
−1 {X (s)} = L
1
−1 {Y ( Y (s)} = L
1
L
L
s2
2 +1
2s 2 s2 + 1
y
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