INTEGRACIÓN:

1) Calcule el cuerpo de revolución que se muestra en la figura se obtiene al girar la curva

dada por , en torno al eje x.

Calcule el volumen utilizando:

a) La regla del trapecio Simple.b) El método de Simpson 1/3 simple.c) El método de Simpson 3/8 simple.

Si el valor exacto es I=11.7286.Calcular el error en cada caso.

Solución:

a) REGLA DEL TRAPECIO SIMPLE.

1)

El volumen:

X=0 X=2

B) EL MÉTODO DE SIMPSON 1/3 SIMPLE.

C) EL MÉTODO DE SIMPSON 3/8 SIMPLE.

2) Repetir el ejercicio anterior pero con los métodos:

a) La regla del trapecio extendido.b) El método de Simpson 1/3 extendido.c) El método de Simpson 3/8 extendido.REPORTAR n, h, integral, error.

Solución:

A) LA REGLA DEL TRAPECIO EXTENDIDO:

n=4:

n=3:

n=2:

n=1:

Reporte:

n h integral error

1 2 15.7080 0.2533

2 1 12.7627 0.0810

3 0.667 12.2007 0.0387

4 0.5 11.9886 0.0218

B) EL MÉTODO DE SIMPSON 3/8 EXTENDIDO (N=6) C) EL MÉTODO DE SIMPSON 1/3 EXTENDIDO (N=4)

n=6:

n=4:

3) Hacer un programa para usar:

a) Trapecio extendido

b) M. Simpson 1/3 extendido

c) Método de Newton-Cotes con n=5

d) Cuadratura de Gauss con n=3

Solución:

Solución a: Solución b:

n=1;a=-1;b=1;i=1;s=0;

f=inline('exp(-(x^2)/2)/(power(2*pi,0.5))');

h=(b-a)/n;

while(i<=n-1)

x=a+h;

s=s+f(x);

i=i+1;

end

y=h/2*(f(a)+2*s+f(b))

a=-1;b=1;

for i=1:11

n=2*(i-1);h=(b-a)/n;x=a;s=0;

if(n>1)

for j=1:n-1

x=x+h;

f=1/sqrt(2*pi)*exp(-x^2/2);

s=f+s;

end

end

fa=1/sqrt(2*pi)*exp(-a^2/2);

fb=1/sqrt(2*pi)*exp(-b^2/2);

s=h/2*(fa+2*s+fb);

end

Solución c:

f=inline('exp(-(x^2)/2)/(power(2*pi,0.5))');

h=(b-a)/n;

while(i<=n-1)

x=a+h;

s=s+f(x);

i=i+1;

end

y=5*h/288*(19*f(a)+75*f(a+h)+50*f(a+2*h)+50*f(a+3*h)+75*f(a+4*h)+19*f(b))

Solución d:

datos: el numero de puntos (2,3,4,5 o 6) por utilizar N, el limite inferor A y limite superior B

Hacer (NP(I);I=1,2,…,5)=(2,3,4,5,6)

Hacer (IAUX(I);I=1,2,…,6)=(1,2,4,6,9,12)

Hacer (Z(I);I=1,2,……,11)=(0.577350269,0.0,0.774596669,0.339981044,0.861136312,0.0,0.538469310,0.906179846, 0.238619186, 0.661209387,0.932469414).

Hacer

(W(I);I=1,2…..,11)=(1,0.0,0.888888888,0.555555555,0.652145155,0.347854845,0.56888888,

0.478628671, 0.23692885, 0.467913955, 0.360761573, 0.171324493).

Hacer I=1

Mientras I<=5,repetir pasos 7 y 8.

Si N=NP(I),ir al paso 10, de otro modo continuar

Hacer I=I+1

Imprimir “N no es 2,3,4,5 o 6”y terminar

Hacer S=0

Hacer I=IAUX(I)

Mientras I<= IAUX(I+1)-1,repetir del 13 al 17.

Hacer ZAUX=(Z(J)*(B-A)+B+A)/2.

Hacer S=S+F(ZAUX*W(J))

Hacer ZAUX=(-Z(J)*(B-A)+B+A)/2.

Hacer S=S+F(ZAUX*W(J))

Hacer J=J+1

Hacer AREA(B-A)/2*S

IMPRIMIR AREA Y TERMINAR

5) Obtenga la primera y segunda derivada en x=1 para la siguiente tabla:

puntos 0 1 2 3 4

X -1 0 2 5 10

F(x) 11 3 23 143 583

Usar Polinomios de Newton (diferencias divididas).

Solución:

P(X) = 11-8(X+1) +6(X+1) * (X)

P(X) = 6X2 - 2X + 3

P1(X) =12X-2

P1 (1) =10

P2(X) =12

P2 (1) =12

6) Resuelva el siguiente:

PVI

Usar: a) Método de Euler b) Método de Euler modificado c) Método de Runge Kutta cuarto orden

Solución:

a) Método de Euler (n=5):

(n=1):

(n=2):

(n=3):

(n=4):

(n=5):

b) SOLUCIÓN, METODO DE EULER MODIFICADO:

c) SOLUCION METODO DE RUNGE KUTTA CUARTO ORDEN:

-Para 5 iteraciones :

(n=1):

(n=2):

(n=3):

(n=4):

(n=5):

7) resuelve el siguiente problema de valor inicial por el método de Runge-Kutta de cuarto orden:

PVI

HACER UN PROGRAMA QUE RESUELVA EL EJERCICIO CON n=8

NOTA: Al escribir la EDO como un sistema, el PVI queda:

PVI

Solución:

Escribiendo del EDO como un sistema:

Buscando el (P.V.I):

y ‘ =z……(f1)

z ‘ = ……..(f2)

y(1) =1

z(1) =2

y(3)= ?

X0=1; y0=1; z0=2 y 8 iteraciones

Hallando h:

Primera iteración:

h=0.25

k1=f1(x0, y0, z0)=f1 (1, 1,2)

c1= f2(x0, y0, z0)=f2 (1, 1, 2)

k1=2

c1=-2

k2=f1(x0+h/2, y0+hk1/2, z0+hc1/2)

c2=f2(x0+h/2, y0+hk1/2, z0+hc1/2)

k2=f1 (1.125,1.25, 1.75)

c2=f2 (1.125,1.25, 1.75)

k2=1.75

c2=-1.8179

k3=f1(x0+h/2, y0+hk2/2, z0+hc2/2)

c3=f2(x0+h/2, y0+hk2/2, z0+hc2/2)

k3=f1 (1.125,1.2188, 1.7728)

c3=f2 (1.125,1.2188,1.7728)

k3=1.7728

c3=-1.8316

k4=f1(x0+h, y0+hk3, z0+hc3)

c4=f2(x0+h, y0+hk3, z0+hc3)

k4=f1 (1.25, 1.4432, 1.5421)

c4=f2 (1.25, 1.4432, 1.5421)

k4=1.5421

c4=-1.7532

x1 = x0+h = 1.25

y1= y0 +h/6 (k1+2*k2+2*k3+k4) = 1.4412

z1= z0 +h/6 (c1+2*c2+2*c3+c4) = 1.5395

Segunda iteración:

k1=f1(x1, y1, z1)= f1 (1.25, 1.4412, 1.5395)

c1= f2(x1, y1, z1)=f2(1.25, 1.4412, 1.5395)

k1=1.5395

c1=-1.7504

k2=f1(x1+h/2, y1+hk1/2, z1+hc1/2)

c2=f2(x1+h/2, y1+hk1/2, z1+hc1/2)

k2=f1 (1.375,1.6336, 1.3207)

c2=f2 (1.375,1.6336, 1.3207)

k2=1.3207

c2=-1.7301

k3=f1(x1+h/2, y1+hk2/2, z1+hc2/2)

c3=f2(x1+h/2, y1+hk2/2, z1+hc2/2)

k3=f1 (1.375,1.6063, 1.3232)

c3=f2 (1.375,1.6063, 1.3232)

k3=1.3232

c3=-1.7190

k4=f1 (x1+h, y1+hk3, z1+hc3)

c4=f2 (x1+h, y1+hk3, z1+hc3)

k4=f1 (1.5, 1.7720, 1.1098)

c4=f2 (1.5, 1.7720, 1.1098)

k4=1.1098

c4=-1.7243

x2 = x1+h = 1.5

y2= y1+h/6 (k1+2*k2+2*k3+k4) = 1.7719

z2= z1 +h/6 (c1+2*c2+2*c3+c4) = 1.1073

Tercera iteración:

k1=f1(x2, y2, z2)=f1 (1.5, 1.7719, 1.1073)

c1= f2(x2, y2, z2)=f2 (1.5, 1.7719, 1.1073)

k1=1.1073

c1=-1.7226

k2=f1 (x2+h/2, y2+hk1/2, z2+hc1/2)

c2=f2 (x2+h/2, y2+hk1/2, z2+hc1/2)

k2=f1 (1.625,1.9103, 0.8920)

c2=f2 (1.625,1.9103, 0.8920)

k2=0.8920

c2=-1.7358

k3=f1 (x2+h/2, y2+hk2/2, z2+hc2/2)

c3=f2(x2+h/2, y2+hk2/2, z2+hc2/2)

k3=f1 (1.625, 1.8834, 0.8903)

c3=f2 (1.625,1.8834, 0.8903)

k3=0.8903

c3=-1.7180

k4=f1 (x2+h, y2+hk3, z2+hc3)

c4=f2 (x2+h, y2+hk3, z2+hc3)

k4=f1 (1.75, 1.9945, 0.6778)

c4=f2 (1.75, 1.9945, 0.6778)

k4=0.6778

c4=-1.7305

x3 = x2+h = 1.75

y3= y2+h/6 (k1+2*k2+2*k3+k4) = 1.9948

z3= z2 +h/6 (c1+2*c2+2*c3+c4) = 0.6756

Cuarta iteración:

k1=f1(x3, y3, z3)=f1 (1.75, 1.9948, 0.6756)

c1= f2(x3, y3, z3)=f2 (1.75, 1.9948, 0.6756)

k1=0.6756

c1=-1.7295

k2=f1 (x3+h/2, y3+hk1/2, z3+hc1/2)

c2=f2 (x3+h/2, y3+hk1/2, z3+hc1/2)

k2=f1 (1.875, 2.0793, 0.4594)

c2=f2 (1.875, 2.0793, 0.4594)

k2=0.4594

c2=-1.7329

k3=f1 (x3+h/2, y3+hk2/2, z3+hc2/2)

c3=f2 (x3+h/2, y3+hk2/2, z3+hc2/2)

k3=f1 (1.875, 2.0522, 0.4590)

c3=f2 (1.875, 2.0522, 0.4590)

k3=0.4590

c3=-1.7133

k4=f1 (x3+h, y3+hk3, z3+hc3)

c4=f2 (x3+h, y3+hk3, z3+hc3)

k4=f1 (2,2.1096, 0.2473)

c4=f2 (2,2.1096, 0.2473)

k4=0.2473

c4=-1.7059

X4 = x3 + h = 2

y4= y3+h/6 (k1+2*k2+2*k3+k4) = 2.1098

z4= z3 +h/6 (c1+2*c2+2*c3+c4) = 0.2453

Quinta iteración:

k1=f1(x4, y4, z4)=f1 (2, 2.1098, 0.2453)

c1= f2(x4, y4, z4)=f2(2, 2.1098, 0.2453)

k1=0.2453

c1=-1.7050

k2=f1 (x4+h/2, y4+hk1/2, z4+hc1/2)

c2=f2 (x4+h/2, y4+hk1/2, z4+hc1/2)

k2=f1 (2.125, 2.1405, 0.0322)

c2=f2 (2.125, 2.1405, 0.0322)

k2=0.0322

c2=-1.6816

k3=f1 (x4+h/2, y4+hk2/2, z4+hc2/2)

c3=f2 (x4+h/2, y4+hk2/2, z4+hc2/2)

k3=f1 (2.125, 2.1138, 0.0351)

c3=f2 (2.125, 2.1138, 0.0351)

k3=0.0351

c3=-1.6622

k4=f1 (x4+h, y4+hk3, z4+hc3)

c4=f2 (x4+h, y4+hk3, z4+hc3)

k4=f1 (2.25, 2.1186, -0.1703)

c4=f2 (2.25, 2.1186, -0.1703)

k4=-0.1703

c4=-1.6244

x5 = x4+h = 2.25

y5= y4+h/6 (k1+2*k2+2*k3+k4) = 2.1185

z5= z4 +h/6 (c1+2*c2+2*c3+c4) = -0.1721

Sexta iteración:

k1=f1(x5, y5, z5)=f1 (2.25, 2.1185, -0.1721)

c1= f2(x5, y5, z5)=f2 (2.25, 2.1185, -0.1721)

k1=-0.1721

c1=-1.6235

k2=f1 (x5+h/2, y5+hk1/2, z5+hc1/2)

c2=f2 (x5+h/2, y5+hk1/2, z5+hc1/2)

k2=f1 (2.375, 2.0970, -0.3750)

c2=f2 (2.375, 2.0970, -0.3750)

k2=-0.3750

c2=-1.5673

k3=f1 (x5+h/2, y5+hk2/2, z5+hc2/2)

c3=f2 (x5+h/2, y5+hk2/2, z5+hc2/2)

k3=f1 (2.375, 2.0716, -0.3680)

c3=f2 (2.375, 2.0716, -0.3680)

k3=-0.3680

c3=-1.5494

k4=f1 (x5+h, y5+hk3, z5+hc3)

c4=f2 (x5+h, y5+hk3, z5+hc3)

k4=f1 (2.5, 2.0265, -0.5595)

c4=f2 (2.5, 2.0265,-0.5595)

k4=-0.5595

c4=-1.4785

x6 = x5+h = 2.5

y6= y5+h/6 (k1+2*k2+2*k3+k4) = 2.0261

z6= z5 +h/6 (c1+2*c2+2*c3+c4) = -0.5611

Sétima iteración:

k1=f1 (x6, y6, z6)=f1 (2.5, 2.0261, -0.5611)

c1= f2 (x6, y6, z6)=f2 (2.5, 2.0261, -0.5611)

k1=-0.5611

c1=-1.4775

k2=f1 (x6+h/2, y6+hk1/2, z6+hc1/2)

c2=f2 (x6+h/2, y6+hk1/2, z6+hc1/2)

k2=f1 (2.625, 1.9560, -0.7458)

c2=f2 (2.625, 1.9560, -0.7458)

k2=-0.7458

c2=-1.3880

k3=f1 (x6+h/2, y6+hk2/2, z6+hc2/2)

c3=f2 (x6+h/2, y6+hk2/2, z6+hc2/2)

k3=f1 (2.625, 1.9329, -0.7346)

c3=f2 (2.625, 1.9329, -0.7346)

k3=-0.7346

c3=-1.3725

k4=f1 (x6+h, y6+hk3, z6+hc3)

c4=f2 (x6+h, y6+hk3, z6+hc3)

k4=f1 (2.75, 1.8425, -0.9042)

c4=f2 (2.75, 1.8425, -0.9042)

k4=-0.9042

c4=-1.2701

x7 = x6+h = 2.75

y7= y6+h/6 (k1+2*k2+2*k3+k4) = 1.8417

z7= z6 +h/6 (c1+2*c2+2*c3+c4) = -0.9056

Octava iteración:

k1=f1(x7, y7, z7)=f1 (2.75, 1.8417, -0.9056)

c1= f2(x7, y7, z7)=f2 (2.75, 1.8417, -0.9056)

k1=-0.9056

c1=-1.2689

k2=f1 (x7+h/2, y7+hk1/2, z7+hc1/2)

c2=f2 (x7+h/2, y7+hk1/2, z7+hc1/2)

k2=f1 (2.875, 1.7285, -1.0642)

c2=f2 (2.875, 1.7285, -1.0642)

k2=-1.0642

c2=-1.1492

k3=f1 (x7+h/2, y7+hk2/2, z7+hc2/2)

c3=f2 (x7+h/2, y7+hk2/2, z7+hc2/2)

k3=f1 (2.875, 1.7089, -1.0493)

c3=f2 (2.875, 1.7089, -1.0493)

k3=-1.0493

c3=-1.1372

k4=f1 (x7+h, y7+hk3, z7+hc3)

c4=f2 (x7+h, y7+hk3, z7+hc3)

k4=f1 (3, 1.5794, -1.1899)

c4=f2 (3, 1.5794, -1.1899)

k4=-1.1899

c4=-1.0073

x8 = x7+h = 3

y8= y7+h/6 (k1+2*k2+2*k3+k4) = 1.5783

z8= z7+h/6 (c1+2*c2+2*c3+c4) = -1.1910

Por lo tanto la respuesta es:

Haciendo el programa que resuelva ejercicio con n=8:

Y (3)= 1.5783

NOTA: al escribir EDO como un sistema, el PVI queda:

y ‘ =z……(f1)

z ‘ = ……..(f2)

y(1) =1

z(1) =2

y(3)=?

x0=1;y0=1;z0=2;xf=3;N=8,I=1;

f1=inline('0*x+0*y+z')

f2=inline('-(z/x)+((1/(x^2))-1)*y')

h=(xf-x0)/N;

while(I<=N)

k1=f1(x0,y0,z0);

c1=f2(x0,y0,z0);

k2=f1(x0+h/2,y0+(h*k1)/2,z0+(h*c1)/2);

c2=f2(x0+h/2,y0+(h*k1)/2,z0+(h*c1)/2);

k3=f1(x0+h/2,y0+(h*k2)/2,z0+(h*c2)/2);

c3=f2(x0+h/2,y0+(h*k2)/2,z0+(h*c2)/2);

k4=f1(x0+h,y0+(h*k3),z0+(h*c3));

c4=f2(x0+h,y0+(h*k3),z0+(h*c3));

x0=x0+h;

y0=y0+(h/6)*(k1+2*k2+2*k3+k4);

z0=z0+(h/6)*(c1+2*c2+2*c3+c4);

I=I+1;

end

y0

8) Resolver:

Solución:

Datos dados en el enunciado:

X0=1; y0=0; z0=1.5

Hallando h:

n=10

Primera iteración:

k1=f1(x0, y0, z0)=f1 (0, 0, 1.5)

c1= f2(x0, y0, z0)=f2 (0, 0, 1.5)

k1=1.5

c1=0

k2=f1(x0+h/2, y0+hk1/2, z0+hc1/2)

c2=f2(x0+h/2, y0+hk1/2, z0+hc1/2)

k2=f1 (0.05, 0.075, 1.5)

c2=f2 (0.05, 0.075, 1.5)

k2=1.5

c2=0.075

k3=f1 (x0+h/2, y0+hk2/2, z0+hc2/2)

c3=f2 (x0+h/2, y0+hk2/2, z0+hc2/2)

k3=f1 (0.05, 0.075, 1.5038)

c3=f2 (0.05, 0.075, 1.5038)

k3=1.5038

c3=0.075

k4=f1 (x0+h, y0+hk3, z0+hc3)

c4=f2 (x0+h, y0+hk3, z0+hc3)

k4=f1 (0.1, 0.1504, 1.5075)

c4=f2 (0.1, 0.1504, 1.5075)

k4=1.5075

c4=0.1504

x1 = x0+h = 0.1

y1= y0 +h/6 (k1+2*k2+2*k3+k4) = 0.1503

z1= z0 +h/6 (c1+2*c2+2*c3+c4) = 1.5075

Segunda iteración:

k1=f1 (x1, y1, z1)=f1 (0.1, 0.1503, 1.5075)

c1= f2 (x1, y1, z1)=f2 (0.1, 0.1503, 1.5075)

k1=1.5075

c1=0.1503

k2=f1 (x1+h/2, y1+hk1/2, z1+hc1/2)

c2=f2 (x1+h/2, y1+hk1/2, z1+hc1/2)

k2=f1 (0.15, 0.2257, 1.5150)

c2=f2 (0.15, 0.2257, 1.5150)

k2=1.5150

c2=0.2257

k3=f1 (x1+h/2, y1+hk2/2, z1+hc2/2)

c3=f2 (x1+h/2, y1+hk2/2, z1+hc2/2)

k3=f1 (0.15, 0.2261, 1.5188)

c3=f2 (0.15, 0.2261, 1.5188)

k3=1.5188

c3=0.2261

k4=f1 (x1+h, y1+hk3, z1+hc3)

c4=f2 (x1+h, y1+hk3, z1+hc3)

k4 =f1 (0.2, 0.3022, 1.5301)

c4=f2 (0.2, 0.3022, 1.5301)

k4=1.5301

c4=0.3022

x2 = x1+h = 0.2

y2= y1+ h/6 (k1+2*k2+2*k3+k4) = 0.3021

z2= z1 + h/6 (c1+2*c2+2*c3+c4) = 1.5301

Tercera iteración:

k1=f1 (x2, y2, z2)=f1 (0.2, 0.3021, 1.5301)

c1= f2 (x2, y2, z2)=f2 (0.2, 0.3021, 1.5301)

k1=1.5301

c1=0.3021

k2=f1 (x2+h/2, y2+hk1/2, z2+hc1/2)

c2=f2 (x2+h/2, y2+hk1/2, z2+hc1/2)

k2=f1 (0.25, 0.3786, 1.5452)

c2=f2 (0.25, 0.3786, 1.5452)

k2=1.5452

c2=0.3786

k3=f1 (x2+h/2, y2+hk2/2, z2+hc2/2)

c3=f2 (x2+h/2, y2+hk2/2, z2+hc2/2)

k3=f1 (0.25, 0.3794, 1.5490)

c3=f2 (0.25, 0.3794, 1.5490)

k3=1.5490

c3=0.3794

k4=f1 (x2+h, y2+hk3, z2+hc3)

c4=f2 (x2+h, y2+hk3, z2+hc3)

k4=f1 (0.3, 0.4570, 1.5680)

c4=f2 (0.3, 0.4570, 1.5680)

k4=1.5680

c4=0.4570

x3 = x2+h = 0.3

y3= y2+h/6 (k1+2*k2+2*k3+k4) = 0.4569

z3= z2 +h/6 (c1+2*c2+2*c3+c4) = 1.5680

Cuarta iteración:

k1=f1(x3, y3, z3)=f1 (0.3, 0.4569, 1.5680)

c1= f2(x3, y3, z3)=f2 (0.3, 0.4569, 1.5680)

k1=1.5680

c1=0.4569

k2=f1 (x3+h/2, y3+hk1/2, z3+hc1/2)

c2=f2 (x3+h/2, y3+hk1/2, z3+hc1/2)

k2=f1 (0.35, 0.5353, 1.5908)

c2=f2 (0.35, 0.5353, 1.5908)

k2=1.5908

c2=0.5353

k3=f1 (x3+h/2, y3+hk2/2, z3+hc2/2)

c3=f2 (x3+h/2, y3+hk2/2, z3+hc2/2)

k3=f1 (0.35, 0.5364, 1.5948)

c3=f2 (0.35, 0.5364, 1.5948)

k3=1.5948

c3=0.5364

k4=f1 (x3+h, y3+hk3, z3+hc3)

c4=f2 (x3+h, y3+hk3, z3+hc3)

k4=f1 (0.4, 0.6164, 1.6216)

c4=f2 (0.4, 0.6164, 1.6216)

k4=1.6216

c4=0.6164

x4 = x3+h = 0.4

y4= y3+h/6 (k1+2*k2+2*k3+k4) = 0.6162

z4= z3 +h/6 (c1+2*c2+2*c3+c4) = 1.6216

Quinta iteración:

k1=f1(x4, y4, z4)=f1 (0.4, 0.6162, 1.6216)

c1= f2(x4, y4, z4)=f2 (0.4, 0.6162, 1.6216)

k1=1.6216

c1=0.6162

k2=f1 (x4+h/2, y4+hk1/2, z4+hc1/2)

c2=f2 (x4+h/2, y4+hk1/2, z4+hc1/2)

k2=f1 (0.45, 0.6973, 1.6524)

c2=f2 (0.45, 0.6973, 1.6524)

k2=1.6524

c2=0.6973

k3=f1 (x4+h/2, y4+hk2/2, z4+hc2/2)

c3=f2(x4+h/2, y4+hk2/2, z4+hc2/2)

k3=f1 (0.45, 0.6988, 1.6565)

c3=f2 (0.45,0.6988, 1.6565)

k3=1.6565

c3=0.6988

k4=f1 (x4+h, y4+hk3, z4+hc3)

c4=f2 (x4+h, y4+hk3, z4+hc3)

k4=f1 (0.5, 0.7819, 1.6915)

c4=f2 (0.5, 0.7819, 1.6915)

k4=1.6915

c4=0.7819

x5 = x4+h = 0.5

y5= y4+h/6 (k1+2*k2+2*k3+k4) = 0.7817

z5= z4 +h/6 (c1+2*c2+2*c3+c4) = 1.6914

Sexta iteración:

k1=f1 (x5, y5, z5)=f1 (0.5, 0.7817, 1.6914)

c1= f2 (x5, y5, z5)=f2 (0.5, 0.7817, 1.6914)

k1=1.6914

c1=0.7817

k2=f1 (x5+h/2, y5+hk1/2, z5+hc1/2)

c2=f2 (x5+h/2, y5+hk1/2, z5+hc1/2)

k2=f1 (0.55, 0.8663, 1.7305)

c2=f2 (0.55, 0.8663, 1.7305)

k2=1.7305

c2=0.8863

k3=f1(x5+h/2, y5+hk2/2, z5+hc2/2)

c3=f2(x5+h/2, y5+hk2/2, z5+hc2/2)

k3=f1 (0.55, 0.8682, 1.7347)

c3=f2 (0.55, 0.8682, 1.7347)

k3=1.7347

c3=0.8682

k4=f1 (x5+h, y5+hk3, z5+hc3)

c4=f2 (x5+h, y5+hk3, z5+hc3)

k4=f1 (0.6, 0.9551, 1.7782)

c4=f2 (0.6, 0.9551, 1.7782)

k4=1.7782

c4=0.9552

x6 = x5+h = 0.6

y6= y5+h/6 (k1+2*k2+2*k3+k4) = 0.9550

z6= z5 +h/6 (c1+2*c2+2*c3+c4) = 1.7788

Sétima iteración:

k1=f1 (x6, y6, z6)=f1 (0.6, 0.9550, 1.7788)

c1= f2 (x6, y6, z6)=f2 (0.6, 0.9550, 1.7788)

k1=1.7788

c1=0.9550

k2=f1 (x6+h/2, y6+hk1/2, z6+hc1/2)

c2=f2 (x6+h/2, y6+hk1/2, z6+hc1/2)

k2=f1 (0.65, 1.0439, 1.8266)

c2=f2 (0.65, 1.0439, 1.8266)

k2=1.8266

c2=1.0439

k3=f1 (x6+h/2, y6+hk2/2, z6+hc2/2)

c3=f2 (x6+h/2, y6+hk2/2, z6+hc2/2)

k3=f1 (0.65, 1.0463, 1.8310)

c3=f2 (0.65, 1.0463, 1.8310)

k3=1.8310

c3=1.0463

k4=f1 (x6+h, y6+hk3, z6+hc3)

c4=f2 (x6+h, y6+hk3, z6+hc3)

k4=f1 (0.70, 1.1381, 1.8834)

c4=f2 (0.70, 1.1381, 1.8834)

k4=1.8834

c4=1.1381

x7 = x6+h = 0.7

y7= y6+h/6 (k1+2*k2+2*k3+k4) = 1.1380

z7= z6 +h/6 (c1+2*c2+2*c3+c4) = 1.8834

Octava iteración:

k1=f1(x7, y7, z7)=f1 (0.70, 1.1380, 1.8834)

c1= f2(x7, y7, z7)=f2 (0.70, 1.1380, 1.8834)

k1=1.8834

c1=1.1380

k2=f1 (x7+h/2, y7+hk1/2, z7+hc1/2)

c2=f2 (x7+h/2, y7+hk1/2, z7+hc1/2)

k2=f1 (0.75, 1.2322, 1.9403)

c2=f2 (0.75, 1.2322, 1.9403)

k2=1.9403

c2=1.2322

k3=f1 (x7+h/2, y7+hk2/2, z7+hc2/2)

c3=f2 (x7+h/2, y7+hk2/2, z7+hc2/2)

k3=f1 (0.75, 1.2350, 1.9450)

c3=f2 (0.75, 1.2350, 1.9450)

k3=1.9450

c3=1.2350

k4=f1 (x7+h, y7+hk3, z7+hc3)

c4=f2 (x7+h, y7+hk3, z7+hc3)

k4=f1 (0.8, 1.3325, 2.0069)

c4=f2 (0.8, 1.3325, 2.0069)

k4=2.0069

c4=1.3325

x8 = x7+h = 0.8

y8= y7+h/6 (k1+2*k2+2*k3+k4) = 1.3323

z8= z7+h/6 (c1+2*c2+2*c3+c4) = 2.0068

Novena iteración:

k1=f1(x8, y8, z8)=f1 (0.80, 1.3323, 2.0068)

c1= f2(x8, y8, z8)=f2(0.80, 1.3323, 2.0068)

k1=2.0068

c1=1.3323

k2=f1 (x8+h/2, y8+hk1/2, z8+hc1/2)

c2=f2 (x8+h/2, y8+hk1/2, z8+hc1/2)

k2=f1 (0.85, 1.4326, 2.0734)

c2=f2 (0.85, 1.4326, 2.0734)

k2=2.0734

c2=1.4326

k3=f1(x8+h/2, y8+hk2/2, z8+hc2/2)

c3=f2(x8+h/2, y8+hk2/2, z8+hc2/2)

k3=f1 (0.85, 1.4360, 2.0784)

c3=f2(0.85, 1.4360, 2.0784)

k3=2.0784

c3=1.4360

k4=f1 (x8+h, y8+hk3, z8+hc3)

c4=f2 (x8+h, y8+hk3, z8+hc3)

k4=f1 (0.9, 1.5401, 2.1504)

c4=f2 (0.9, 1.5401, 2.1504)

k4=2.1504

c4=1.5401

x9 = x8+h = 0.9

y9= y8+h/6 (k1+2*k2+2*k3+k4) = 1.5400

z9= z8+h/6 (c1+2*c2+2*c3+c4) = 2.1503

Decima iteración:

k1=f1 (x9, y9, z9)=f1 (0.9, 1.5400, 2.1503)

c1= f2 (x9, y9, z9)=f2 (0.9, 1.5400, 2.1503)

k1=2.1503

c1=1.5400

k2=f1(x9+h/2, y9+hk1/2, z9+hc1/2)

c2=f2(x9+h/2, y9+hk1/2, z9+hc1/2)

k2=f1 (0.95, 1.6475, 2.2273)

c2=f2 (0.95, 1.6475, 2.2273)

k2=2.2273

c2=1.6475

k3=f1 (x9+h/2, y9+hk2/2, z9+hc2/2)

c3=f2 (x9+h/2, y9+hk2/2, z9+hc2/2)

k3=f1 (0.95, 1.6514, 2.2327)

c3=f2 (0.95, 1.6514, 2.2327)

k3=2.2327

c3=1.6514

k4=f1 (x9+h, y9+hk9, z9+hc3)

c4=f2 (x9+h, y9+hk9, z9+hc3)

k4=f1 (1, 1.7633, 2.3154)

c4=f2 (1, 1.7633, 2.3154)

k4=2.3154

c4=1.7633

x10 = x9+h = 1

y10= y9+h/6 (k1+2*k2+2*k3+k4) = 1.7633

z10= z9+h/6 (c1+2*c2+2*c3+c4) = 2.3153

Por lo tanto la respuesta es

BIBLIOGRAFIA:

MÉTODOS NÚMERICOS APLICADOS ALA INGENIERIA: Antonio Nieves, Federico C. Domínguez.

2ªedición, CAPITULOS:

CAPITULO 6: Integración y diferenciación

CAPITULO7: Ecuaciones diferenciales ordinarias.

Y (1)= 1.7631