METODO DE RUNGE KUTTA

K1= f(Xi, Yi)

K2= f(Xi + h/2, Yi + h*K1/2)

K3= f(Xi + h/2, Yi + h*K2/2)

K4= f(Xi + h, Yi + h*K3)

yi+1=Yi + (h/6)*( K1 + 2* K2 + 2* K3 + K4 )

EJEMPLO

F(x,y)= x/y.^2; y(0)=1; con h=0.2; y(1)=?

I=0; x=0; y=1;

K1= f(0,1)= 0/1^2=0

K2= f(0+0.2/2, 1+0.2*0/2)= f(0.1,1)= 0.1/1^2= 0.1

K3= f(0+0.2/2, 1+0.2*0.1/2)= f(0.1,1.01)= 0.1/(1.01)^2 = 0.0980

K4= f(0 + 0.2, 1 + 0.2*0.098)= f(0.2, 1.0196)= 0.2/(1.0196)^2 = 0.1924

Y1= 1 + (0.2/6)*(0+ 2*0.1 + 2*0.098 + 0.1924)= 1.0196

I=1; x=0.2; y=1.0196;

K1= f(0.2,1.0196)= 0.2/1.0196^2= 0.19238461655668

K2= f(0.2+0.2/2, 1.0196+0.2*0.1924/2)= f(0.3, 1.03884)= 0.3/1.03884^2= 0.27798664211471

K3= f(0.2+0.2/2, 1.0196+0.2*0.27798664211471/2)= f(0.3, 1.04739866421147)= 0.3/(1.04739866421147)^2 = 0.27346214960243

K4= f(0.2 + 0.2, 1.0196 + 0.2*0.27346214960243)= f(0.4, 1.07429242992049)= 0.4/(1.07429242992049)^2 = 0.34658914796485

Y2=
1.0196 + (0.2/6)*( 0.19238461655668+ 2*0.27798664211471+ 2*0.27346214960243+ 0.34658914796485)= 1.07432955771330

I=2; x=0.4; y=1.07432955771330;

K1= f(0.4, 1.07432955771330)= 0.4/1.07432955771330^2= 0.34656519280578

K2= f(0.4+0.2/2, 1.07432955771330+0.2*0.34656519280578/2)= f(0.50, 1.10898607699388)= 0.50/1.10898607699388^2= 0.40655360542654

K3= f(0.4+0.2/2, 1.07432955771330+0.2*0.40655360542654/2)= f(0.5, 1.11498491825595)= 0.5/(1.11498491825595)^2 = 0.40219069476747

K4= f(0.4 + 0.2, 1.07432955771330+ 0.2*0.40219069476747)= f(0.6, 1.15476769666679)= 0.6/( 1.15476769666679)^2 = 0.44994765986148

Y3= 1.07432955771330 + (0.2/6)*( 0.34656519280578+ 2*0.40655360542654+ 2*0.40219069476747 + 0.44994765986148)= 1.15479627281514

I=3; x=0.6; y=1.15479627281514;

K1= f(0.4, 1.07432955771330)= 0.4/1.07432955771330^2= 0.34656519280578

K2= f(0.4+0.2/2, 1.07432955771330+0.2*0.34656519280578/2)= f(0.50, 1.10898607699388)= 0.50/1.10898607699388^2= 0.40655360542654

K3= f(0.4+0.2/2, 1.07432955771330+0.2*0.40655360542654/2)= f(0.5, 1.11498491825595)= 0.5/(1.11498491825595)^2 = 0.40219069476747

K4= f(0.4 + 0.2, 1.07432955771330+ 0.2*0.40219069476747)= f(0.6, 1.15476769666679)= 0.6/( 1.15476769666679)^2 = 0.44994765986148

Y3= 1.07432955771330 + (0.2/6)*( 0.34656519280578+ 2*0.40655360542654+ 2*0.40219069476747 + 0.44994765986148)= 1.15479627281514

K1= f(Xi, Yi)

K2= f(Xi + h/2, Yi + h*K1/2)

K3= f(Xi + h/2, Yi + h*K2/2)

K4= f(Xi + h, Yi + h*K3)

yi+1=Yi + (h/6)*( K1 + 2* K2 + 2* K3 + K4 )