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