INTEGRACIÓN: 1) Calcule el cuerpo cuerpo de revolución revolución que que se muestra muestra en la figura figura se obtiene obtiene al girar la curva dada por
, en torno al eje x.
Calcule el volumen utilizando: a) La regla del trapecio trapecio Simple. Simple. b) El métod método o de Simpson Simpson 1/3 simple. c) El méto método do de Simp Simpso son n 3/8 3/8 simple. X=0 X
Si el valor exacto es I=11.7286. Calcular el error en cada caso.
Solución: a) REGLA DEL TRAPEC T RAPECIO IO SIMPLE SIM PLE. 1) El volumen:
B) EL MÉTODO DE SIMPSON 1/3 SIMPLE.
C) EL MÉTODO DE SIMPSON 3/8 SIMPLE.
2) Repetir el ejercicio ejercicio anterior anterior pero con los métodos: 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 TRAPECI TRAPECIO O EXTEN EXTENDIDO: DIDO:
n=4:
n=3:
n=2:
n=1:
Reporte: n
h
integral
error
1 2 3 4
2 1 0.667 0.5
15.7080 12.7627 12.2007 11.9886
B) EL MÉTODO DE SIMPSON 3/8 EXTENDIDO (N=6) C) EL MÉTODO DE SIMPSON 1/3 EXTENDIDO (N=4)
n=6:
n=4:
0.2533 0.0810 0.0387 0.0218
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;
a=-1;b=1;
f=inline('exp(-(x^2)/2)/ (power(2*pi,0.5))');
for i=1:11
h=(b-a)/n;
n=2*(i-1);h=(b-a)/n;x=a;s=0; if(n>1)
while(i<=n-1)
for j=1:n-1
x=a+h;
x=x+h;
s=s+f(x); i=i+1;
f=1/sqrt(2*pi)*exp(x^2/2);
end
s=f+s;
y=h/2*(f(a)+2*s+f(b))
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)=(1,0.0,0.888888888,0.555555555,0.652145155,0.347854 845,0.56888888, 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 y segunda derivada en x=1 para la siguiente tabla: 5) Obtenga la primera Hacer AREA(B-A)/2*S puntos
0
1
2
3
4
X
-1
0
2
5
10
F(x)
11
3
23
143
583
IMPRIMIR AREA Y TERMINAR
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 RungeKutta 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: h=0.2 5
Primera iteración: 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: Y (3)= 1.5783 Haciendo el programa que resuelva ejercicio con n=8: 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 Y (1)= 1.7631
BIBLIOGRAFIA:
MÉTODOS NÚMERICOS APLICADOS ALA INGENIERIA: Antonio Nieves, Federico C. Domínguez. 2ªedición, CAPITULOS:
CAPITULO 6: Integración y diferenciación CAPITULO7: Ecuaciones diferenciales ordinarias.
