Ralston

El metodo de Ralston calcula la pendiente en un punto inicial (xi,yi)

m=k1=f (xi,yi)

 

 y con el estima un nuevo punto;

(xi+0.75*h , yi+0.75*k1h)

Con este punto calcula una nueva pendiente

m=k2=f (xi+0.75*h , yi+0.75k1h)

 

 y con esta dos pendiente calcula el yi+1

yi+1=yi+(k1+2*k2)*h/3

Ejemplo

(dy/dx)=yx^2-1.1y                donde y(0)=1                    para x=[0,1]

Con un h=0.25 y realiza nuevamente tus calculos pero con h=0.05

h= 0.25          
xi yi k1 Xi+0.75h yi+0.75k1h k2 yi+1=(k1+2k2)*h/3
0 1.0000 -1.1 0.1875 0.79375 -0.8452197 0.76746338
0.25 0.7675 -0.7962433 0.4375 0.61816777 -0.5616634 0.60749921
0.5 0.6075 -0.5163743 0.6875 0.51067903 -0.3203713 0.5110728
0.75 0.5111 -0.2747016 0.9375 0.45956625 -0.1016072 0.47124646
1 0.4712          
             
h= 0.05          
xi yi k1 Xi+0.75h yi+0.75k1h k2 yi+1=(k1+2k2)*h/3
0 1 -1.1 0.0375 0.95875 -1.0532768 0.94655744
0.05 0.9466 -1.0388468 0.0875 0.90760069 -0.9914119 0.89619626
0.1 0.8962 -0.9768539 0.1375 0.85956424 -0.9292695 0.84893971
0.15 0.8489 -0.9147325 0.1875 0.81463724 -0.8674614 0.80477879
0.2 0.8048 -0.8530655 0.2375 0.77278884 -0.8064776 0.76367845
0.25 0.7637 -0.7923164 0.2875 0.73396658 -0.7466963 0.7255833
0.3 0.7256 -0.7328391 0.3375 0.69810183 -0.6883939 0.69042285
0.35 0.6904 -0.6748883 0.3875 0.66511454 -0.6317549 0.65811621
0.4 0.6581 -0.6186292 0.4375 0.63491762 -0.5768822 0.62857632
0.45 0.6286 -0.5641472 0.4875 0.6074208 -0.5238055 0.60171368
0.5 0.6017 -0.5114566 0.5375 0.58253406 -0.4724897 0.57743975
0.55 0.5774 -0.4605082 0.5875 0.56017069 -0.4228413 0.5556699
0.6 0.5557 -0.4111957 0.6375 0.54025006 -0.3747141 0.53632617
0.65 0.5363 -0.363361 0.6875 0.52270013 -0.3279127 0.51933973
0.7 0.5193 -0.3167972 0.7375 0.50745983 -0.2821952 0.50465327
0.75 0.5047 -0.2712511 0.7875 0.49448135 -0.2372738 0.49222329
0.8 0.4922 -0.2264227 0.8375 0.48373244 -0.1928127 0.48202249
0.85 0.4820 -0.1819635 0.8875 0.47519886 -0.1484254 0.47404225
0.9 0.4740 -0.1374723 0.9375 0.46888704 -0.103668 0.46829545
0.95 0.4683 -0.0924884 0.9875 0.46482713 -0.0580308 0.46481961
1 0.4648          

 

Grafica que compara la solucion analitica con la obtenida por el metodod con h=0.25 u h=0.05.