Ecuaciones Diferenciales Ordinarias

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Ecuaciones Diferenciales OrdinariasSolucin Numrica

Ecuacion Diferencial : Ecuaciones que involucran variables dependientes y sus derivadas con respecto a las variables independientes son llamadas ecuaciones diferenciales. Ecuacion Diferencial Ordinaria : ecuaciones diferenciales que involucran solamente UNA variable independiente son llamadas ecuaciones diferenciales ordinarias.Ecuacin Diferencial Parcial: : ecuaciones diferenciales que involucran dos o mas variables independiente son llamadas ecuaciones diferenciales parciales.EDO- Ecuacin Diferencial OrdinariaSoluciones de EDOs Analtica y NumricaResolver la EDO para encontrar una familia de soluciones.Elije la solucin que satisface las condiciones iniciales correctas. Encuentra una frmula analtica parar y(t)Empieza con las condic. inicialesResuelve para pequeos tamaos de paso (t).Resuelve aprox. en cada t Encuentra pares de puntos: (t0,y0), (t1,y1),

Mtodo de Solucin AnalticaMtodo de Solucin NumricaLa solucin analtica de la ecuacin diferencial ordinaria as como ecuaciones diferenciales parciales se llama la " solucin de la forma cerrada Esta solucin requiere que las constantes de la integracin estn evaluadas usando valores prescritos de variable(s) independiente(s).

Ecuaciones Diferenciales OrdinariasEcuaciones Diferenciales OrdinariasEn el mejor de los casos, solamente algunas ecuaciones diferenciales se puede solucionar analticamente en una forma cerrada.

En la mayora de los problemas prcticos de la ingeniera que implican ecuaciones diferenciales requieren el uso de mtodos numricos. Metodos de un solo pasoEl objetivo consiste en solucionar una EDO en forma discreta, obteniendo un nuevo punto a partir de un punto anterioryxyihyi+1xixi+1y(x)(xi+1, yi+1)=(xi ,yi, h)Mtodo de Taylor de orden k

Sea una EDO de primer orden:Podemos usar la serie de Taylor para aproximar la solucin de la EDO, haciendo:

Mtodo de Taylor de orden k

Podemos plantear el algoritmo siguiente:

Siendo E el error de truncamiento.


Ejemplo:- Estime y(x) para x=0.1, 0.2, 0.3, 0.4 y 0.5, usando Taylor de orden 3 Solucin




Metodo de EulerPermite resolver una EDO de primer orden de la forma: Valor nuevo = Valor anterior + (Tamao del paso) x (Pendiente) yxyihyi+1

xixi+1Metodo de EulerLa primera derivada proporciona un estimado directo de la pendiente en xiLa ecuacin es aplicada iterativamente, un paso a la vez, sobre una distancia pequea para reducir el errorPor esto se conoce como mtodo de un solo paso.EJEMPL0

Para la condicin inicial y(1)=1, determine y para h = 0.1 analticamente y usando el mtodo de Euler:

dy/dxC.I..Tamao del pasoRecordar la solucin analtica fue 1.4413.Si reducimos el tamao del paso a 0,05 y aplicamos Euler dos vecesRecordar la solucin analtica es 1.4413

Obtenemos:Anlisis del Error -Mtodo de EulerError de truncacin - causado por la naturaleza de la tcnica empleada para aproximar los valores de yError local de truncacin (a partir de la Serie de Taylor)Propagacin del error de truncacinSuma de los dos es el error globalError de Redondeo causado por el numero limitados de dgitos significativos que pueden ser retenidos por computadora o calculadoraMtodo de Euler Ejemplo

solucin Analtica

Mtodo de Euler Ejemplontnynfn= - yn+1yn+1= yn+Dt fn000.0001.0000.10010.10.1000.9000.19020.20.1900.8100.27130.30.2710.7290.34440.40.3440.6560.41050.50.4100.5900.46960.60.4690.5310.52270.70.5220.4780.57080.80.5700.4300.61390.90.6130.3870.651Mtodo de Euler Mejorado o HeunUn error fundamental en el mtodo de Euler es que se asume la derivada en el principio del intervalo para aplicarse a travs de todo el intervalo. Una simple modificacin ser demostrada.Esta modificacin pertenece realmente a una clase ms grande de las tcnicas de solucin llamadas Runge-Kutta que exploremos ms adelante. Mtodo de HeunConsidere la siguiente expansin de Taylor:

Aproxime f con una diferencia progresiva

Substituyendo en la expansin

Mtodo de HeunDetermine las derivadas para el intervaloPunto inicialPunto final (basado en el paso de Euler a partir del punto inicial)Use el promedio para obtener una estimacin mejorada de la pendiente para el intervalo completo Podemos pensar en el paso de Euler como paso de prueba.Mtodo de Heunyxi xi+1Evaluar la pendiente en xiLa proyeccin consigue f(xi+1 )Basado en el tamao del paso hhyxi xi+1hyxi xi+1Ahora determine la pendiente en xi+1yxi xi+1Tomar los promedios de estas dos pendientesyxi xi+1yxi xi+1Use esta pendiente promedio para predecir yi+1

{yxi xi+1

{Use esta pendiente promedio para predecir yi+1yxi xi+1yxxi xi+1

yxxi xi+1

Metodo de Euler Mejorado (Heun)Permite resolver una EDO de primer orden de la forma:

Metodo de Euler Mejorado (Heun)Ejemplo

Metodo de Euler Mejorado (Heun)Ejemplo

Metodo de Runge-Kutta de orden 2A partir del mtodo de Heun podemos deducir el mtodo de Runge-Kutta

Metodo de Runge-Kutta de orden 2Ejemplo

Se obtienen los mismos resultados que el mtodo de Euler MejoradoMetodo de Runge-Kutta de orden 4

Metodo de Runge-Kutta de orden 4

Los mtodos para solucionar una ecuacion diferencial de primer orden pueden ser adaptados a la solucin de sistemas de primer orden.

Sistemas de Ecuaciones Diferenciales de Primer OrdenPor ejemplo sea el siguiente sistema de dos ecuaciones diferenciales ordinarias de primer orden:

Sistemas de Ecuaciones Diferenciales de Primer OrdenDonde busca aproximar y(x) y z(x)Resolver el siguiente Problema de Valor Inicial que consta de dos EDOs de primer orden:

Sistemas de Ecuaciones Diferenciales de Primer OrdenDonde busca aproximar y(1.2) y z(1.2)

Sistemas de Ecuaciones Diferenciales de Primer OrdenPlantearemos el algoritmo para el mtodo de Euler:

Sistemas de Ecuaciones Diferenciales de Primer OrdenReemplanzado valores:Sistemas de Ecuaciones Diferenciales de Primer OrdenSe tiene una solucin aproximada en forma discreta:n xn yn zn0 1 1 2 1 1.1 1.4 2.22 1.2 1.87 2.401

Sistemas de Ecuaciones Diferenciales de Primer OrdenSi queremos mejorar la exactitud del resultado podemos usar un paso h mas pequeo o usar Taylor, por ejemplo de orden 2 sera:

Sistemas de Ecuaciones Diferenciales de Primer OrdenTambin se puede hacer una adaptacin del mtodo de Runge-Kutta 2Los problemas de valor inicial de mayor orden pueden ser transformados en un sistema de ecuaciones diferenciales de primer orden.

Ecuaciones Diferenciales orden SuperiorPor ejemplo, sea la EDO de tercer orden:Ecuaciones Diferenciales orden Superior

La EDO de tercer orden se transforma en un sistema de 3 ecuaciones de primer orden:Ecuaciones Diferenciales orden Superior

Considere una ecuacin diferencial de segundo orden de un sistema de masa y resorte vibratorio

Las cond. iniciales son x(0) =x0 y x(0) =0.

Ecuaciones Diferenciales orden SuperiorRe-escribir la ecuacin:

La primera derivada puede ser escrita:

Ecuaciones Diferenciales orden SuperiorLa ecuacin puede ser escrita como un conjunto de dos ecuaciones de primer orden.

Las condiciones iniciales: x(0) = x0 y v(0) = 0.

Ecuaciones Diferenciales orden SuperiorSistemas de Valor Inicial Problemas Las ecuaciones pueden ser definidas:

Podemos aplicar Euler:

Sistemas de Valor Inicial Problemas Diferenciales mayor-orden Problemas EjemploConsidere una ecuacin diferencial de segundo orden para sistemas de masa-resorte vibrante.

Las condiciones iniciales son x(0) =0.2, x(0) =0 y Dt = 0.02. (Solucin Exacta = 0.2 cos(2t))

Problema EjemploLa ecuacin puede ser escrita como un conjunto de dos ecuaciones de primer orden.

Las condiciones iniciales, x(0) = 0.2 y v(0) = 0.

Problema EjemploEl desarrollo del mtodo de Euler.

Problema EjemploEjemplo

Se puede observar un error que cada vez se ir incrementando.

Las ecuaciones son definidas como funciones.

Las condiciones iniciales, x(0) = 0.2 and v(0) = 0.

Problema EjemploLos componentes de Runge-Kutta:

ki,j donde i es el paso y j es la funcin.

Problema EjemploLa actualizacin de un slo paso:

Use los valores iniciales x(0) = 0.02 y v(0) = 0

Problema EjemploEjemplo Metodo de Runge-Kutta de 4th Orden

Los puntos tienen menos error que el mtodo de Euler.

La aproximacin depende del tamao del paso del problema

Ejemplo Metodo de Runge-Kutta de 4th OrdenSistemas de EDO - Problema Valor InicialEstas tcnicas pueden trabajar con grandes sistemas de ecuaciones para realizar una serie de integracines del problema. Las ecuaciones se pueden solucionar como serie de EDOs.

Dando un conjunto de valores iniciales, y1,y2,y1 e y2.Sistemas de EDO - Problema Valor Inicial

El problema es formado por 4 EDOs de primer orden con cuatro variables y condiciones iniciales.Sistemas de EDO - Problema Valor Inicial

El problema puede ser escrito en el formato matricial y solucionado por consiguiente .Sistemas de EDO - Problema Valor InicialFuerzas pueden ser aadidas y fijadas para solucionar las ecuaciones.

Sistemas de EDO - Problema Valor InicialCuando las condiciones la EDO se dan por lo menos en algn punto diferente del valor inicial de la variable independiente.Sistemas de EDO - Problema Valor Frontera

Condiciones de Frontera Condiciones Inicialesy(0)=0 y(L)=0 y(0)=0 y(0)=073Mtodo de Diferencias FinitasSea la ecuacin diferencial ordinaria de segundo orden:

Dividiendo el intervalo en (n+1) partes iguales

74Mtodo de Diferencias FinitasSean las frmulas de diferenciacin numrica para la primera y segunda derivada

75Mtodo de Diferencias FinitasReemplazando en la ecuacin diferencial para cada nodo i=1, 2, , n:

76Mtodo de Diferencias FinitasSe tendr un sistema de n ecuaciones con n incgnitas:

77Mtodo de Diferencias FinitasAgrupando:

78Mtodo de Diferencias FinitasLuego:

79Mtodo de Diferencias FinitasExpresado en forma matricial tenemos un sistema tridiagonal:

80Mtodo de Diferencias FinitasEjemplo.- Resolver la siguiente ecuacion diferencial ordinaria:y-y-2y=0 con condiciones de frontera: y(0)=0.1 e y(0.5)=0.283. considere h=0.1.Solucion.-Discretizacin:x0x1x2x3x4X500.10.20.30.40.5y0y1y2y3y4y50.1????????0.28381Mtodo de Diferencias FinitasSe usarn las siguientes frmulas de diferenciacin numrica:

Sea la ecuacin diferencial para cada nodo i:

82Mtodo de Diferencias FinitasReemplazando para cada nodo:

83Mtodo de Diferencias FinitasTeniendo en cuenta que: y0=0.1, y5=0.283 y h=0.1

84Mtodo de Diferencias FinitasPlanteando y resolviendo el sistema tridiagonal:

85Mtodo del DisparoSea la ecuacion diferencial de segundo orden con condiciones de frontera:

Consiste en transformar el problema de valor frontera en un problema de valor inicial, suponiendo una pendiente s, luego se desarrolla un mtodo numrico para encontrar uN(s), se compara con B, si estos valores no son aproximados se sigue suponiendo pendientes hasta dar en el blanco B.86Mtodo del DisparoEl problema de valor inicial resultante:

87Mtodo del Disparo

88Mtodo del Disparo

89Mtodo de DisparoEjemplo.- Resolver la siguiente ecuacion diferencial ordinaria:y-y-2y=0 con condiciones de frontera: y(0)=0.1 e y(0.5)=0.283. considere h=0.1.Solucin.-

Luego debemos resolver el Problema de Valor Inicial:90Mtodo de Disparo

Mediante un cambio de variable tendremos un sistema de dos ecuaciones diferenciales de primer orden:El cual lo resolvemos por Runge-Kutta de orden 4, como se puede ver en la siguiente tabla:Mtodo de DisparoResultados mediante Runge-Kutta de orden 4:ixiyizi=yi00.00.10.3660010.10.139660.4295220.20.186430.5087630.30.242040.6070640.40.308610.7284950.50.388670.87803

Mtodo de DisparoCalculando una nueva pendiente aproximada s1:ixiyizi=yi00.00.10.1546610.10.117360.1936920.20.139010.2409030.30.165870.2981540.40.199050.3677050.50.239910.45232

Mtodo de DisparoMediante interpolacin lineal obtenemos la tercera pendiente s3:ixiyizi=yi00.00.10.2158810.10.123820.2620020.20.152740.3184930.30.187930.3876340.40.230780.4722150.50.283000.57564

Sheet1CVEN 302 Assignment 5Problem 5The Dx = 0.1 is not required however if you want to check your solutions using it. The spreadsheet works.Problem 12. 5Euler Methodf =y *x -2Dx =0.02Dx =0.1txvdx/dtdv/dtValor exactoxyfactual value00.200-0.80.212220.020.2-0.016-0.016-0.80.19984002131.12.21.81818181822.19033887970.040.19968-0.032-0.032-0.798720.19936034131.22.38181818181.6540404042.36272082570.060.19904-0.0479744-0.0479744-0.796160.19856172721.32.54722222221.50723208422.51913707250.080.198080512-0.0638976-0.0638976-0.7923220480.19744545671.42.69794543061.37650277072.66142439490.10.19680256-0.079744041-0.079744041-0.787210240.19601331560.120.1952076792-0.0954882458-0.0954882458-0.78083071670.1942675950.140.1932979143-0.1111048601-0.1111048601-0.77319165710.19221108770.160.1910758171-0.1265686932-0.1265686932-0.76430326830.18984708360.180.1885444432-0.1418547586-0.1418547586-0.75417777280.18717936470.20.185707348-0.1569383141-0.1569383141-0.74282939210.18421219880.220.1825685817-0.1717949019-0.1717949019-0.7302743270.18095033260.240.1791326837-0.1864003884-0.1864003884-0.71653073480.17739898460.260.1754046759-0.2007310031-0.2007310031-0.70161870380.17356383590.280.1713900559-0.2147633772-0.2147633772-0.68556022350.16945102220.30.1670947883-0.2284745817-0.2284745817-0.66837915330.1650671230.320.1625252967-0.2418421647-0.2418421647-0.65010118680.16041915160.340.1576884534-0.2548441885-0.2548441885-0.63075381360.15551454380.360.1525915696-0.2674592648-0.2674592648-0.61036627850.15036114580.380.1472423843-0.2796665903-0.2796665903-0.58896953740.14496720210.40.1416490525-0.2914459811-0.2914459811-0.56659621010.13934134190.420.1358201329-0.3027779053-0.3027779053-0.54328053160.13349256520.440.1297645748-0.3136435159-0.3136435159-0.51905829920.12743022880.460.1234917045-0.3240246819-0.3240246819-0.49396681790.12116403130.480.1170112108-0.3339040183-0.3339040183-0.46804484340.11470399720.50.1103331305-0.3432649151-0.3432649151-0.44133252190.10806046120.520.1034678322-0.3520915656-0.3520915656-0.41387132870.10124405140.540.0964260009-0.3603689921-0.3603689921-0.38570400350.09426567280.560.089218621-0.3680830722-0.3680830722-0.35687448410.08713648930.580.0818569596-0.3752205619-0.3752205619-0.32742783830.07986790590.60.0743525483-0.3817691187-0.3817691187-0.29741019340.07247155090.620.066717166-0.3877173225-0.3877173225-0.26686866390.06495925690.640.0589628195-0.3930546958-0.3930546958-0.23585127810.05734304190.660.0511017256-0.3977717214-0.3977717214-0.20440690240.04963509030.680.0431462912-0.4018598594-0.4018598594-0.17258516470.04184773320.70.035109094-0.4053115627-0.4053115627-0.1404363760.03399342860.720.0270028627-0.4081202902-0.4081202902-0.1080114510.02608474170.740.0188404569-0.4102805192-0.4102805192-0.07536182770.01813432490.760.0106348465-0.4117877558-0.4117877558-0.04253938620.0101548970.780.0023990914-0.4126385435-0.4126385435-0.00959636570.00215922340.8-0.0058536794-0.4128304708-0.41283047080.0234147178-0.00583990450.82-0.0141102889-0.4123621765-0.41236217650.0564411554-0.01382968970.84-0.0223575324-0.4112333534-0.41123335340.0894301295-0.02179735040.86-0.0305821995-0.4094447508-0.40944475080.1223287978-0.02973014010.88-0.0387710945-0.4069981748-0.40699817480.1550843779-0.03761536780.9-0.046911058-0.4038964873-0.40389648730.1876442319-0.04544041890.92-0.0549889877-0.4001436026-0.40014360260.2199559508-0.05319277510.94-0.0629918598-0.3957444836-0.39574448360.251967439-0.06086003420.96-0.0709067494-0.3907051348-0.39070513480.2836269977-0.06842993020.98-0.0787208521-0.3850325949-0.38503259490.3148834085-0.0758903531-0.086421504-0.3787349267-0.37873492670.3456860161-0.08322936731.02-0.0939962026-0.3718212064-0.37182120640.3759848102-0.09043523241.04-0.1014326267-0.3643015102-0.36430151020.4057305068-0.09749642051.06-0.1087186569-0.3561869-0.35618690.4348746276-0.1044016351.08-0.1158423949-0.3474894075-0.34748940750.4633695796-0.11113982931.1-0.122792183-0.3382220159-0.33822201590.4911687322-0.11770022351.12-0.1295566234-0.3283986413-0.32839864130.5182264934-0.12407232241.14-0.1361245962-0.3180341114-0.31803411140.5444983847-0.13024593211.16-0.1424852784-0.3071441437-0.30714414370.5699411137-0.13621117611.18-0.1486281613-0.2957453214-0.29574532140.5945126452-0.14195851131.2-0.1545430677-0.2838550685-0.28385506850.6181722709-0.14747874311.22-0.1602201691-0.2714916231-0.27149162310.6408806764-0.15276304041.24-0.1656500015-0.2586740096-0.25867400960.6626000062-0.15780294941.26-0.1708234817-0.2454220094-0.24542200940.683293927-0.16259040741.28-0.1757319219-0.2317561309-0.23175613090.7029276877-0.16711775541.3-0.1803670445-0.2176975772-0.21769757720.7214681782-0.17137775071.32-0.1847209961-0.2032682136-0.20326821360.7388839844-0.17536357811.34-0.1887863604-0.1884905339-0.18849053390.7551454415-0.17906886121.36-0.192556171-0.1733876251-0.17338762510.7702246842-0.18248767231.38-0.1960239235-0.1579831314-0.15798313140.7840956942-0.1856145421.4-0.1991835862-0.1423012175-0.14230121750.7967343447-0.18844446811.42-0.2020296105-0.1263665306-0.12636653060.8081184421-0.19097292331.44-0.2045569411-0.1102041618-0.11020416180.8182277645-0.19319586251.46-0.2067610244-0.0938396065-0.09383960650.8270440975-0.19510972951.48-0.2086378165-0.0772987245-0.0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2ndf =y *x -2f ' =( y' *x2 - y* 2x )/( x4 )Dx =0.25xyff 'actual value122-221.252.43751.56-1.49762.44280551631.52.78071.2358666667-1.09854814812.79122485021.753.0553370370.9976610733-0.81441720273.070126018523.27930176780.8198254419-0.61486908153.2974425414Midpoint Methodf =y *x -2Dx =0.25Dx =0.1xyff2actual valuexyff2actual value1221.777777777821221.904761904821.252.44444444441.56444444441.39636363642.44280551631221.904761904821.52.79353535351.24157126821.11667948392.79122485021221.904761904821.753.07270522451.00333231820.90968796853.07012601851221.9047619048223.30012721660.82503180420.75366227093.29744254141221.904761904821221.904761904821221.904761904821221.904761904821221.904761904821221.904761904821221.90476190482Runge Kutta Classicf =y *x -2Dx =0.25xyfk1k2k3k4actual value1220.50.44444444440.4389574760.390233196221.252.44283950621.5634172840.3908543210.3488617080.34608533690.30988053812.44280551631.52.79127766431.24056785080.31014196270.27894425050.27746743570.25050980412.79122485021.753.07019018751.00251108160.25062777040.22723584520.22640413230.2060371453.070126018523.29751433250.82437858310.20609464580.18826638920.18777287350.17211294843.2974425414Dx =0.1xyfk1k2k3k4actual value1220.20.19047619050.19004427170.180995394421.12.19033938641.810197840.1810197840.17246497380.17214154050.16406117552.19033887971.22.36272171781.64077897070.16407789710.15646468260.15622105980.14904986852.36272082571.32.51913825951.49061435480.14906143550.14231379850.14212867810.13577892542.51913707251.42.66142581191.35787031220.13578703120.12981304770.12967097910.12404874632.66142439491.52.79122645041.24054508910.12405450890.11876186080.1186517120.11366711572.79122485021.62.90998457871.13671272610.11367127260.10897411260.1088878470.10445925352.90998282921.73.01896031961.0446229480.10446229480.1002838030.10021558270.09627086123.01895844591.83.11924897420.96273116490.09627311650.09254596150.09249151070.08896788053.11924699521.93.21180163110.88969574270.08896957430.08563540880.0855915670.082434833.211799561423.29744469040.82436117260.08243611730.07944468170.07940909060.07657264813.2974425414Predictor Corrector SectionDx =0.25Runge Kutta Resultsxyfk1k2k3k4actual value1220.50.44444444440.4389574760.390233196221.252.44283950621.5634172840.3908543210.3488617080.34608533690.30988053812.44280551631.52.79127766431.24056785080.31014196270.27894425050.27746743570.25050980412.79122485021.753.07019018751.00251108160.25062777040.22723584520.22640413230.2060371453.070126018523.29751433250.82437858310.20609464580.18826638920.18777287350.17211294843.2974425414Predictor-Corrector - (2nd order)Dx =0.25xyfy*f*1221.252.44283950621.5634172842.77912098771.23516488341.52.79266227711.24118323433.06267882941.00005839331.753.07281748051.00336897323.29393294120.823483235323.30117400660.82529350173.48523794810.68844206382.25Predictor-Corrector - (3rd order)Dx =0.25xyfy*f*1221.252.44283950621.5634172841.52.79127766431.24056785083.07291066481.00339940081.753.06998855031.00244524093.2996602450.824915061323.29714624580.82428656143.48719429390.68882850252.25Predictor-Corrector - (4th order)Dx =0.25xyfy*f*1221.252.44283950621.5634172841.52.79127766431.24056785081.753.07019018751.00251108163.29718024790.82429506223.29755418890.82438854723.48529868070.68845406042.25

Sheet1

xvValor exactoTime (t)DisplacementEuler Example

Sheet2

Sheet3

Chart10.200.20.2-0.0160.19984002130.19968-0.0320.19936034130.19904-0.04797440.19856172720.198080512-0.06389760.19744545670.19680256-0.0797440410.19601331560.1952076792-0.09548824580.1942675950.1932979143-0.11110486010.19221108770.1910758171-0.12656869320.18984708360.1885444432-0.14185475860.18717936470.185707348-0.15693831410.18421219880.1825685817-0.17179490190.18095033260.1791326837-0.18640038840.17739898460.1754046759-0.20073100310.17356383590.1713900559-0.21476337720.16945102220.1670947883-0.22847458170.1650671230.1625252967-0.24184216470.16041915160.1576884534-0.25484418850.15551454380.1525915696-0.26745926480.15036114580.1472423843-0.27966659030.14496720210.1416490525-0.29144598110.13934134190.1358201329-0.30277790530.13349256520.1297645748-0.31364351590.12743022880.1234917045-0.32402468190.12116403130.1170112108-0.33390401830.11470399720.1103331305-0.34326491510.10806046120.1034678322-0.35209156560.10124405140.0964260009-0.36036899210.09426567280.089218621-0.36808307220.08713648930.0818569596-0.37522056190.07986790590.0743525483-0.38176911870.07247155090.066717166-0.38771732250.06495925690.0589628195-0.39305469580.05734304190.0511017256-0.39777172140.04963509030.0431462912-0.40185985940.04184773320.035109094-0.40531156270.03399342860.0270028627-0.40812029020.02608474170.0188404569-0.41028051920.01813432490.0106348465-0.41178775580.0101548970.0023990914-0.41263854350.0021592234-0.0058536794-0.4128304708-0.0058399045-0.0141102889-0.4123621765-0.0138296897-0.0223575324-0.4112333534-0.0217973504-0.0305821995-0.4094447508-0.0297301401-0.0387710945-0.4069981748-0.0376153678-0.046911058-0.4038964873-0.0454404189-0.0549889877-0.4001436026-0.0531927751-0.0629918598-0.3957444836-0.0608600342-0.0709067494-0.3907051348-0.0684299302-0.0787208521-0.3850325949-0.075890353-0.086421504-0.3787349267-0.0832293673-0.0939962026-0.3718212064-0.0904352324-0.1014326267-0.3643015102-0.0974964205-0.1087186569-0.3561869-0.104401635-0.1158423949-0.3474894075-0.1111398293-0.122792183-0.3382220159-0.1177002235-0.1295566234-0.3283986413-0.1240723224-0.1361245962-0.3180341114-0.1302459321-0.1424852784-0.3071441437-0.1362111761-0.1486281613-0.2957453214-0.1419585113-0.1545430677-0.2838550685-0.1474787431-0.1602201691-0.2714916231-0.1527630404-0.1656500015-0.2586740096-0.1578029494-0.1708234817-0.2454220094-0.1625904074-0.1757319219-0.2317561309-0.1671177554-0.1803670445-0.2176975772-0.1713777507-0.1847209961-0.2032682136-0.1753635781-0.1887863604-0.1884905339-0.1790688612-0.192556171-0.1733876251-0.1824876723-0.1960239235-0.1579831314-0.185614542-0.1991835862-0.1423012175-0.1884444681-0.2020296105-0.1263665306-0.1909729233-0.2045569411-0.1102041618-0.1931958625-0.2067610244-0.0938396065-0.1951097295-0.2086378165-0.0772987245-0.1967114626-0.210183791-0.0606076992-0.1979984993-0.211395945-0.0437929959-0.1989687807-0.2122718049-0.0268813203-0.1996207544-0.2128094313-0.0098995759-0.1999533775-0.21300742280.0071251786-0.1999661179-0.21286491920.0241657724-0.1996589552-0.21238160380.0411949659-0.1990323807-0.21155770450.0581854942-0.1980873968-0.21039399460.0751101106-0.1968255154-0.20889179240.0919416302-0.1952487551-0.20705295980.1086529735-0.1933596385-0.20487990030.1252172103-0.1911611877-0.20237555610.1416076024-0.1886569198-0.19954340410.1577976468-0.1858508411-0.19638745110.1737611192-0.1827474406-0.19291222870.1894721153-0.1793516833-0.18912278640.2049050936-0.1756690015-0.18502468460.2200349165-0.1717052867-0.18062398620.2348368912-0.1674668802-0.17592724840.2492868101-0.1629605624-0.17094151220.26336099-0.1581935424-0.16567429240.277036311-0.1531734465-0.16013356620.2902902544-0.1479083058-0.15432776110.3031009397-0.1424065433-0.14826574230.3154471606-0.1366769607-0.14195679910.3273084199-0.1307287242-0.13541063070.3386649639-0.1245713496-0.12863733140.3494978143-0.1182146874-0.12164737510.3597888008-0.1116689069-0.11445159910.3695205908-0.10494448

xvactual valueTime (t)DisplacementEuler Example

Sheet1CVEN 302 Assignment 5Problem 5Problem 12. 5Euler Methodf =0.020.1txvdx/dtdv/dtactual valuexyfactual value00.200-0.80.212220.020.2-0.016-0.016-0.80.19984002131.12.21.81818181822.19033887970.040.19968-0.032-0.032-0.798720.19936034131.22.38181818181.6540404042.36272082570.060.19904-0.0479744-0.0479744-0.796160.19856172721.32.54722222221.50723208422.51913707250.080.198080512-0.0638976-0.0638976-0.7923220480.19744545671.42.69794543061.37650277072.66142439490.10.19680256-0.079744041-0.079744041-0.787210240.19601331560.120.1952076792-0.0954882458-0.0954882458-0.78083071670.1942675950.140.1932979143-0.1111048601-0.1111048601-0.77319165710.19221108770.160.1910758171-0.1265686932-0.1265686932-0.76430326830.18984708360.180.1885444432-0.1418547586-0.1418547586-0.75417777280.18717936470.20.185707348-0.1569383141-0.1569383141-0.74282939210.18421219880.220.1825685817-0.1717949019-0.1717949019-0.7302743270.18095033260.240.1791326837-0.1864003884-0.1864003884-0.71653073480.17739898460.260.1754046759-0.2007310031-0.2007310031-0.70161870380.17356383590.280.1713900559-0.2147633772-0.2147633772-0.68556022350.16945102220.30.1670947883-0.2284745817-0.2284745817-0.66837915330.1650671230.320.1625252967-0.2418421647-0.2418421647-0.65010118680.16041915160.340.1576884534-0.2548441885-0.2548441885-0.63075381360.15551454380.360.1525915696-0.2674592648-0.2674592648-0.61036627850.15036114580.380.1472423843-0.2796665903-0.2796665903-0.58896953740.14496720210.40.1416490525-0.2914459811-0.2914459811-0.56659621010.13934134190.420.1358201329-0.3027779053-0.3027779053-0.54328053160.13349256520.440.1297645748-0.3136435159-0.3136435159-0.51905829920.12743022880.460.1234917045-0.3240246819-0.3240246819-0.49396681790.12116403130.480.1170112108-0.3339040183-0.3339040183-0.46804484340.11470399720.50.1103331305-0.3432649151-0.3432649151-0.44133252190.10806046120.520.1034678322-0.3520915656-0.3520915656-0.41387132870.10124405140.540.0964260009-0.3603689921-0.3603689921-0.38570400350.09426567280.560.089218621-0.3680830722-0.3680830722-0.35687448410.08713648930.580.0818569596-0.3752205619-0.3752205619-0.32742783830.07986790590.60.0743525483-0.3817691187-0.3817691187-0.29741019340.07247155090.620.066717166-0.3877173225-0.3877173225-0.26686866390.06495925690.640.0589628195-0.3930546958-0.3930546958-0.23585127810.05734304190.660.0511017256-0.3977717214-0.3977717214-0.20440690240.04963509030.680.0431462912-0.4018598594-0.4018598594-0.17258516470.04184773320.70.035109094-0.4053115627-0.4053115627-0.1404363760.03399342860.720.0270028627-0.4081202902-0.4081202902-0.1080114510.02608474170.740.0188404569-0.4102805192-0.4102805192-0.07536182770.01813432490.760.0106348465-0.4117877558-0.4117877558-0.04253938620.0101548970.780.0023990914-0.4126385435-0.4126385435-0.00959636570.00215922340.8-0.0058536794-0.4128304708-0.41283047080.0234147178-0.00583990450.82-0.0141102889-0.4123621765-0.41236217650.0564411554-0.01382968970.84-0.0223575324-0.4112333534-0.41123335340.0894301295-0.02179735040.86-0.0305821995-0.4094447508-0.40944475080.1223287978-0.02973014010.88-0.0387710945-0.4069981748-0.40699817480.1550843779-0.03761536780.9-0.046911058-0.4038964873-0.40389648730.1876442319-0.04544041890.92-0.0549889877-0.4001436026-0.40014360260.2199559508-0.05319277510.94-0.0629918598-0.3957444836-0.39574448360.251967439-0.06086003420.96-0.0709067494-0.3907051348-0.39070513480.2836269977-0.06842993020.98-0.0787208521-0.3850325949-0.38503259490.3148834085-0.0758903531-0.086421504-0.3787349267-0.37873492670.3456860161-0.08322936731.02-0.0939962026-0.3718212064-0.37182120640.3759848102-0.09043523241.04-0.1014326267-0.3643015102-0.36430151020.4057305068-0.09749642051.06-0.1087186569-0.3561869-0.35618690.4348746276-0.1044016351.08-0.1158423949-0.3474894075-0.34748940750.4633695796-0.11113982931.1-0.122792183-0.3382220159-0.33822201590.4911687322-0.11770022351.12-0.1295566234-0.3283986413-0.32839864130.5182264934-0.12407232241.14-0.1361245962-0.3180341114-0.31803411140.5444983847-0.13024593211.16-0.1424852784-0.3071441437-0.30714414370.5699411137-0.13621117611.18-0.1486281613-0.2957453214-0.29574532140.5945126452-0.14195851131.2-0.1545430677-0.2838550685-0.28385506850.6181722709-0.14747874311.22-0.1602201691-0.2714916231-0.27149162310.6408806764-0.15276304041.24-0.1656500015-0.2586740096-0.25867400960.6626000062-0.15780294941.26-0.1708234817-0.2454220094-0.24542200940.683293927-0.16259040741.28-0.1757319219-0.2317561309-0.23175613090.7029276877-0.16711775541.3-0.1803670445-0.2176975772-0.21769757720.7214681782-0.17137775071.32-0.1847209961-0.2032682136-0.20326821360.7388839844-0.17536357811.34-0.1887863604-0.1884905339-0.18849053390.7551454415-0.17906886121.36-0.192556171-0.1733876251-0.17338762510.7702246842-0.18248767231.38-0.1960239235-0.1579831314-0.15798313140.7840956942-0.1856145421.4-0.1991835862-0.1423012175-0.14230121750.7967343447-0.18844446811.42-0.2020296105-0.1263665306-0.12636653060.8081184421-0.19097292331.44-0.2045569411-0.1102041618-0.11020416180.8182277645-0.19319586251.46-0.2067610244-0.0938396065-0.09383960650.8270440975-0.19510972951.48-0.2086378165-0.0772987245-0.0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=f ' =0.25xyff 'actual value122-221.252.43751.56-1.49762.44280551631.52.78071.2358666667-1.09854814812.79122485021.753.0553370370.9976610733-0.81441720273.070126018523.27930176780.8198254419-0.61486908153.2974425414Midpoint Methodf =0.250.1xyff2actual valuexyff2actual value1221.777777777821221.904761904821.252.44444444441.56444444441.39636363642.44280551631221.904761904821.52.79353535351.24157126821.11667948392.79122485021221.904761904821.753.07270522451.00333231820.90968796853.07012601851221.9047619048223.30012721660.82503180420.75366227093.29744254141221.904761904821221.904761904821221.904761904821221.904761904821221.904761904821221.904761904821221.90476190482Runge Kutta Classicf =0.25xyfactual value1220.50.44444444440.4389574760.390233196221.252.44283950621.5634172840.3908543210.3488617080.34608533690.30988053812.44280551631.52.79127766431.24056785080.31014196270.27894425050.27746743570.25050980412.79122485021.753.07019018751.00251108160.25062777040.22723584520.22640413230.2060371453.070126018523.29751433250.82437858310.20609464580.18826638920.18777287350.17211294843.29744254140.1xyfactual value1220.20.19047619050.19004427170.180995394421.12.19033938641.810197840.1810197840.17246497380.17214154050.16406117552.19033887971.22.36272171781.64077897070.16407789710.15646468260.15622105980.14904986852.36272082571.32.51913825951.49061435480.14906143550.14231379850.14212867810.13577892542.51913707251.42.66142581191.35787031220.13578703120.12981304770.12967097910.12404874632.66142439491.52.79122645041.24054508910.12405450890.11876186080.1186517120.11366711572.79122485021.62.90998457871.13671272610.11367127260.10897411260.1088878470.10445925352.90998282921.73.01896031961.0446229480.10446229480.1002838030.10021558270.09627086123.01895844591.83.11924897420.96273116490.09627311650.09254596150.09249151070.08896788053.11924699521.93.21180163110.88969574270.08896957430.08563540880.0855915670.082434833.211799561423.29744469040.82436117260.08243611730.07944468170.07940909060.07657264813.2974425414Predictor Corrector Section0.25Runge Kutta Resultsxyfactual value1220.50.44444444440.4389574760.390233196221.252.44283950621.5634172840.3908543210.3488617080.34608533690.30988053812.44280551631.52.79127766431.24056785080.31014196270.27894425050.27746743570.25050980412.79122485021.753.07019018751.00251108160.25062777040.22723584520.22640413230.2060371453.070126018523.29751433250.82437858310.20609464580.18826638920.18777287350.17211294843.29744254140.25xyfy*f*1221.252.44283950621.5634172842.77912098771.23516488341.52.79266227711.24118323433.06267882941.00005839331.753.07281748051.00336897323.29393294120.823483235323.30117400660.82529350173.48523794810.68844206382.250.25xyfy*f*1221.252.44283950621.5634172841.52.79127766431.24056785083.07291066481.00339940081.753.06998855031.00244524093.2996602450.824915061323.29714624580.82428656143.48719429390.68882850252.250.25xyfy*f*1221.252.44283950621.5634172841.52.79127766431.24056785081.753.07019018751.00251108163.29718024790.82429506223.29755418890.82438854723.48529868070.68845406042.25

Sheet1


Sheet2

Sheet3

Sheet1CVEN 302 Assignment 5Problem 5The Dx = 0.1 is not required however if you want to check your solutions using it. The spreadsheet works.Problem 12. 5Euler Methodf =y *x -2Dx =0.02txvdx/dtdv/dtactual value00.200-0.80.20.020.2-0.016-0.016-0.80.19984002130.040.19968-0.032-0.032-0.798720.19936034130.060.19904-0.0479744-0.0479744-0.796160.19856172720.080.198080512-0.0638976-0.0638976-0.7923220480.19744545670.10.19680256-0.079744041-0.079744041-0.787210240.19601331560.120.1952076792-0.0954882458-0.0954882458-0.78083071670.1942675950.140.1932979143-0.1111048601-0.1111048601-0.77319165710.19221108770.160.1910758171-0.1265686932-0.1265686932-0.76430326830.18984708360.180.1885444432-0.1418547586-0.1418547586-0.75417777280.18717936470.20.185707348-0.1569383141-0.1569383141-0.74282939210.18421219880.220.1825685817-0.1717949019-0.1717949019-0.7302743270.18095033260.240.1791326837-0.1864003884-0.1864003884-0.71653073480.17739898460.260.1754046759-0.2007310031-0.2007310031-0.70161870380.17356383590.280.1713900559-0.2147633772-0.2147633772-0.68556022350.16945102220.30.1670947883-0.2284745817-0.2284745817-0.66837915330.1650671230.320.1625252967-0.2418421647-0.2418421647-0.65010118680.16041915160.340.1576884534-0.2548441885-0.2548441885-0.63075381360.15551454380.360.1525915696-0.2674592648-0.2674592648-0.61036627850.15036114580.380.1472423843-0.2796665903-0.2796665903-0.58896953740.14496720210.40.1416490525-0.2914459811-0.2914459811-0.56659621010.13934134190.420.1358201329-0.3027779053-0.3027779053-0.54328053160.13349256520.440.1297645748-0.3136435159-0.3136435159-0.51905829920.12743022880.460.1234917045-0.3240246819-0.3240246819-0.49396681790.12116403130.480.1170112108-0.3339040183-0.3339040183-0.46804484340.11470399720.50.1103331305-0.3432649151-0.3432649151-0.44133252190.10806046120.520.1034678322-0.3520915656-0.3520915656-0.41387132870.10124405140.540.0964260009-0.3603689921-0.3603689921-0.38570400350.09426567280.560.089218621-0.3680830722-0.3680830722-0.35687448410.08713648930.580.0818569596-0.3752205619-0.3752205619-0.32742783830.07986790590.60.0743525483-0.3817691187-0.3817691187-0.29741019340.07247155090.620.066717166-0.3877173225-0.3877173225-0.26686866390.06495925690.640.0589628195-0.3930546958-0.3930546958-0.23585127810.05734304190.660.0511017256-0.3977717214-0.3977717214-0.20440690240.04963509030.680.0431462912-0.4018598594-0.4018598594-0.17258516470.04184773320.70.035109094-0.4053115627-0.4053115627-0.1404363760.03399342860.720.0270028627-0.4081202902-0.4081202902-0.1080114510.02608474170.740.0188404569-0.4102805192-0.4102805192-0.07536182770.01813432490.760.0106348465-0.4117877558-0.4117877558-0.04253938620.0101548970.780.0023990914-0.4126385435-0.4126385435-0.00959636570.00215922340.8-0.0058536794-0.4128304708-0.41283047080.0234147178-0.00583990450.82-0.0141102889-0.4123621765-0.41236217650.0564411554-0.01382968970.84-0.0223575324-0.4112333534-0.41123335340.0894301295-0.02179735040.86-0.0305821995-0.4094447508-0.40944475080.1223287978-0.02973014010.88-0.0387710945-0.4069981748-0.40699817480.1550843779-0.03761536780.9-0.046911058-0.4038964873-0.40389648730.1876442319-0.04544041890.92-0.0549889877-0.4001436026-0.40014360260.2199559508-0.05319277510.94-0.0629918598-0.3957444836-0.39574448360.251967439-0.06086003420.96-0.0709067494-0.3907051348-0.39070513480.2836269977-0.06842993020.98-0.0787208521-0.3850325949-0.38503259490.3148834085-0.0758903531-0.086421504-0.3787349267-0.37873492670.3456860161-0.08322936731.02-0.0939962026-0.3718212064-0.37182120640.3759848102-0.09043523241.04-0.1014326267-0.3643015102-0.36430151020.4057305068-0.09749642051.06-0.1087186569-0.3561869-0.35618690.4348746276-0.1044016351.08-0.1158423949-0.3474894075-0.34748940750.4633695796-0.11113982931.1-0.122792183-0.3382220159-0.33822201590.4911687322-0.11770022351.12-0.1295566234-0.3283986413-0.32839864130.5182264934-0.12407232241.14-0.1361245962-0.3180341114-0.31803411140.5444983847-0.13024593211.16-0.1424852784-0.3071441437-0.30714414370.5699411137-0.13621117611.18-0.1486281613-0.2957453214-0.29574532140.5945126452-0.14195851131.2-0.1545430677-0.2838550685-0.28385506850.6181722709-0.14747874311.22-0.1602201691-0.2714916231-0.27149162310.6408806764-0.15276304041.24-0.1656500015-0.2586740096-0.25867400960.6626000062-0.15780294941.26-0.1708234817-0.2454220094-0.24542200940.683293927-0.16259040741.28-0.1757319219-0.2317561309-0.23175613090.7029276877-0.16711775541.3-0.1803670445-0.2176975772-0.21769757720.7214681782-0.17137775071.32-0.1847209961-0.2032682136-0.20326821360.7388839844-0.17536357811.34-0.1887863604-0.1884905339-0.18849053390.7551454415-0.17906886121.36-0.192556171-0.1733876251-0.17338762510.7702246842-0.18248767231.38-0.1960239235-0.1579831314-0.15798313140.7840956942-0.1856145421.4-0.1991835862-0.1423012175-0.14230121750.7967343447-0.18844446811.42-0.2020296105-0.1263665306-0.12636653060.8081184421-0.19097292331.44-0.2045569411-0.1102041618-0.11020416180.8182277645-0.19319586251.46-0.2067610244-0.0938396065-0.09383960650.8270440975-0.19510972951.48-0.2086378165-0.0772987245-0.07729872450.834551266-0.19671146261.5-0.210183791-0.0606076992-0.06060769920.840735164-0.19799849931.52-0.211395945-0.0437929959-0.04379299590.84558377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Kutta Classicf =y *x -2Dx 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Sheet1


Sheet2

vxExactoTime (t)Displacement4th order Runge Kutta Example

Sheet3

Chart200.20.2-0.01599573330.19984002130.1998400213-0.03196587690.19936034130.1993603413-0.04788488190.19856172720.1985617272-0.06372728130.19744545680.1974454567-0.07946773060.19601331570.1960133156-0.09508104860.19426759520.194267595-0.11054225710.1922110880.1922110877-0.12582662160.1898470840.1898470836-0.14090969040.18717936520.1871793647-0.15576733370.18421219940.1842121988-0.17037578260.18095033340.1809503326-0.18471166650.17739898540.1773989846-0.19875205120.1735638370.1735638359-0.2124744750.16945102340.1694510222-0.2258569850.16506712440.165067123-0.23887817210.16041915310.1604191516-0.2515172050.15551454550.1555145438-0.2637538640.15036114780.1503611458-0.27556857320.14496720430.1449672021-0.28694243140.13934134420.1393413419-0.2978572430.13349256780.1334925652-0.30829554660.12743023170.1274302288-0.31824064310.12116403440.1211640313-0.32767662240.11470400050.1147039972-0.33658838910.10806046470.1080604612-0.34496168620.10124405520.1012440514-0.35278311810.09426567680.0942656728-0.36004017240.08713649350.0871364893-0.36672123930.07986791040.0798679059-0.37281563040.07247155560.0724715509-0.3783135960.06495926180.0649592569-0.38320634060.05734304710.0573430419-0.38748603690.04963509570.0496350903-0.39114583820.04184773880.0418477332-0.39417988960.03399343440.0339934286-0.39658333730.02608474780.0260847417-0.39835233630.01813433120.0181343249-0.39948405650.01015490340.010154897-0.39997668750.00215923010.0021592234-0.3998294412-0.0058398976-0.0058399045-0.399042553-0.0138296827-0.0138296897-0.397617282-0.0217973433-0.0217973504-0.3955559081-0.0297301328-0.0297301401-0.3928617292-0.0376153604-0.0376153678-0.3895390553-0.0454404114-0.0454404189-0.3855932021-0.0531927675-0.0531927751-0.3810304821-0.0608600265-0.0608600342-0.3758581946-0.0684299224-0.0684299302-0.3700846141-0.0758903451-0.075890353-0.3637189773-0.0832293594-0.0832293673-0.3567714677-0.0904352245-0.0904352324-0.3492531999-0.0974964126-0.0974964205-0.3411762015-0.1044016272-0.104401635-0.332553394-0.1111398214-0.1111398293-0.3233985721-0.1177002157-0.1177002235-0.3137263814-0.1240723147-0.1240723224-0.3035522954-0.1302459245-0.1302459321-0.2928925906-0.1362111686-0.1362111761-0.2817643201-0.1419585039-0.1419585113-0.2701852869-0.1474787359-0.1474787431-0.2581740148-0.1527630334-0.1527630404-0.2457497194-0.1578029427-0.1578029494-0.2329322769-0.1625904009-0.1625904074-0.2197421924-0.1671177491-0.1671177554-0.2062005673-0.1713777446-0.1713777507-0.1923290654-0.1753635723-0.1753635781-0.1781498779-0.1790688557-0.1790688612-0.1636856887-0.1824876672-0.1824876723-0.1489596373-0.1856145373-0.185614542-0.1339952823-0.1884444638-0.1884444681-0.1188165634-0.1909729193-0.1909729233-0.1034477634-0.1931958589-0.1931958625-0.087913469-0.1951097264-0.1951097295-0.0722385318-0.1967114599-0.1967114626-0.0564480284-0.1979984971-0.1979984993-0.0405672202-0.1989687789-0.1989687807-0.0246215131-0.1996207532-0.1996207544-0.0086364169-0.1999533768-0.19995337750.0073624958-0.1999661177-0.19996611790.0233496301-0.1996589555-0.19965895520.0392994099-0.1990323816-0.19903238070.0551863191-0.1980873983-0.19808739680.0709849419-0.1968255174-0.19682551540.086670004-0.1952487578-0.19524875510.1022164125-0.1933596418-0.19335963850.1175992966-0.1911611916-0.19116118770.132794047-0.1886569243-0.18865691980.1477763552-0.1858508461-0.18585084110.1625222527-0.1827474463-0.18274744060.1770081493-0.1793516896-0.17935168330.1912108707-0.1756690084-0.17566900150.2051076955-0.1717052943-0.17170528670.2186763917-0.1674668884-0.16746688020.2318952523-0.1629605712-0.16296056240.2447431301-0.1581935519-0.15819354240.257199471-0.1531734566-0.15317344650.2692443477-0.1479083165-0.14790830580.2808584909-0.1424065546-0.14240654330.2920233204-0.1366769727-0.13667696070.302720975-0.1307287367-0.13072872420.3129343405-0.1245713627-0.12457134960.3226470779-0.118214701-0.11821468740.3318436489-0.1116689211-0.11166890690.3405093408-0.1049444948-0.104944480.3486302904-0.0980521796-0.09805216430.3561935059-0.0910030019-0.0910029860.3631868878-0.0838082387-0.08380822240.3695992483-0.0764794003-0.07647938350.3754203288-0.0690282112-0.0690281940.3806408168-0.0614665917-0.0614665740.3852523608-0.0538066387-0.05380662060.3892475833-0.0460606066-0.04606058810.3926200926-0.0382408873-0.03824086850.3953644936-0.0303599908-0.03035997170.3974763958-0.0224305248-0.02243050540.3989524205-0.0144651747-0.01446515510.3997902065-0.0064766835-0.00647666360.39998841350.00152216910.00152218920.39954672440.00951858650.00951860680.39846584580.01749977620.01749979670.39674750680.025452970.02545299060.39439445650.03336544450.03336546520.39141045930.04122454140.04122456210.38780028880.04901768790.04901770850.38356972060.05673241640.05673243710.37872552270.06435638530.06435640580.37327544480.07187739760.07187741810.36722820580.07928342150.07928344180.36059348010.08656260870.0865626289

vxactual valueTime (t)Displacement4th order Runge Kutta Example

Sheet1CVEN 302 Assignment 5Problem 5Problem 12. 5Euler Methodf =0.02txvdx/dtdv/dtactual value00.200-0.80.20.020.2-0.016-0.016-0.80.19984002130.040.19968-0.032-0.032-0.798720.19936034130.060.19904-0.0479744-0.0479744-0.796160.19856172720.080.198080512-0.0638976-0.0638976-0.7923220480.19744545670.10.19680256-0.079744041-0.079744041-0.787210240.19601331560.120.1952076792-0.0954882458-0.0954882458-0.78083071670.1942675950.140.1932979143-0.1111048601-0.1111048601-0.77319165710.19221108770.160.1910758171-0.1265686932-0.1265686932-0.76430326830.18984708360.180.1885444432-0.1418547586-0.1418547586-0.75417777280.18717936470.20.185707348-0.1569383141-0.1569383141-0.74282939210.18421219880.220.1825685817-0.1717949019-0.1717949019-0.7302743270.18095033260.240.1791326837-0.1864003884-0.1864003884-0.71653073480.17739898460.260.1754046759-0.2007310031-0.2007310031-0.70161870380.17356383590.280.1713900559-0.2147633772-0.2147633772-0.68556022350.16945102220.30.1670947883-0.2284745817-0.2284745817-0.66837915330.1650671230.320.1625252967-0.2418421647-0.2418421647-0.65010118680.16041915160.340.1576884534-0.2548441885-0.2548441885-0.63075381360.15551454380.360.1525915696-0.2674592648-0.2674592648-0.61036627850.15036114580.380.1472423843-0.2796665903-0.2796665903-0.58896953740.14496720210.40.1416490525-0.2914459811-0.2914459811-0.56659621010.13934134190.420.1358201329-0.3027779053-0.3027779053-0.54328053160.13349256520.440.1297645748-0.3136435159-0.3136435159-0.51905829920.12743022880.460.1234917045-0.3240246819-0.3240246819-0.49396681790.12116403130.480.1170112108-0.3339040183-0.3339040183-0.46804484340.11470399720.50.1103331305-0.3432649151-0.3432649151-0.44133252190.10806046120.520.1034678322-0.3520915656-0.3520915656-0.41387132870.10124405140.540.0964260009-0.3603689921-0.3603689921-0.38570400350.09426567280.560.089218621-0.3680830722-0.3680830722-0.35687448410.08713648930.580.0818569596-0.3752205619-0.3752205619-0.32742783830.07986790590.60.0743525483-0.3817691187-0.3817691187-0.29741019340.07247155090.620.066717166-0.3877173225-0.3877173225-0.26686866390.06495925690.640.0589628195-0.3930546958-0.3930546958-0.23585127810.05734304190.660.0511017256-0.3977717214-0.3977717214-0.20440690240.04963509030.680.0431462912-0.4018598594-0.4018598594-0.17258516470.04184773320.70.035109094-0.4053115627-0.4053115627-0.1404363760.03399342860.720.0270028627-0.4081202902-0.4081202902-0.1080114510.02608474170.740.0188404569-0.4102805192-0.4102805192-0.07536182770.01813432490.760.0106348465-0.4117877558-0.4117877558-0.04253938620.0101548970.780.0023990914-0.4126385435-0.4126385435-0.00959636570.00215922340.8-0.0058536794-0.4128304708-0.41283047080.0234147178-0.00583990450.82-0.0141102889-0.4123621765-0.41236217650.0564411554-0.01382968970.84-0.0223575324-0.4112333534-0.41123335340.0894301295-0.02179735040.86-0.0305821995-0.4094447508-0.40944475080.1223287978-0.02973014010.88-0.0387710945-0.4069981748-0.40699817480.1550843779-0.03761536780.9-0.046911058-0.4038964873-0.40389648730.1876442319-0.04544041890.92-0.0549889877-0.4001436026-0.40014360260.2199559508-0.05319277510.94-0.0629918598-0.3957444836-0.39574448360.251967439-0.06086003420.96-0.0709067494-0.3907051348-0.39070513480.2836269977-0.06842993020.98-0.0787208521-0.3850325949-0.38503259490.3148834085-0.0758903531-0.086421504-0.3787349267-0.37873492670.3456860161-0.08322936731.02-0.0939962026-0.3718212064-0.37182120640.3759848102-0.09043523241.04-0.1014326267-0.3643015102-0.36430151020.4057305068-0.09749642051.06-0.1087186569-0.3561869-0.35618690.4348746276-0.1044016351.08-0.1158423949-0.3474894075-0.34748940750.4633695796-0.11113982931.1-0.122792183-0.3382220159-0.33822201590.4911687322-0.11770022351.12-0.1295566234-0.3283986413-0.32839864130.5182264934-0.12407232241.14-0.1361245962-0.3180341114-0.31803411140.5444983847-0.13024593211.16-0.1424852784-0.3071441437-0.30714414370.5699411137-0.13621117611.18-0.1486281613-0.2957453214-0.29574532140.5945126452-0.14195851131.2-0.1545430677-0.2838550685-0.28385506850.6181722709-0.14747874311.22-0.1602201691-0.2714916231-0.27149162310.6408806764-0.15276304041.24-0.1656500015-0.2586740096-0.25867400960.6626000062-0.15780294941.26-0.1708234817-0.2454220094-0.24542200940.683293927-0.16259040741.28-0.1757319219-0.2317561309-0.23175613090.7029276877-0.16711775541.3-0.1803670445-0.2176975772-0.21769757720.7214681782-0.17137775071.32-0.1847209961-0.2032682136-0.20326821360.7388839844-0.17536357811.34-0.1887863604-0.1884905339-0.18849053390.7551454415-0.17906886121.36-0.192556171-0.1733876251-0.17338762510.7702246842-0.18248767231.38-0.1960239235-0.1579831314-0.15798313140.7840956942-0.1856145421.4-0.1991835862-0.1423012175-0.14230121750.7967343447-0.18844446811.42-0.2020296105-0.1263665306-0.12636653060.8081184421-0.19097292331.44-0.2045569411-0.1102041618-0.11020416180.8182277645-0.19319586251.46-0.2067610244-0.0938396065-0.09383960650.8270440975-0.19510972951.48-0.2086378165-0.0772987245-0.07729872450.834551266-0.19671146261.5-0.210183791-0.0606076992-0.06060769920.840735164-0.19799849931.52-0.211395945-0.0437929959-0.04379299590.84558377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Kutta Classicf =0.02txvactual 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Ecuaciones Diferenciales Ordinarias

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