!Runge-Kutta methods
!first order
clc
clear all
close all
format long
x1=[1000];
y1=[1000];
x2=[1000];
y2=[1000];
x3=[1000];
y3=[1000];
x4=[1000];
y4=[1000];
x1(1)=input('enter  x0 = ' );
y1(1)=input ('enter  yo = ');
k=input('enter reng  ');
h=input('enter h ');
for t=1:1:k
x1(t+1)=x1(1)+t*h;
y1(t+1)=y1(t)+h*(x1(t)+y1(t));
end
y1(k)
!*********************************************************************************!
!second order
x2(1)=x1(1);
y2(1)=y1(1);
for t=1:1:k
    k1=h*(x2(t)+y2(t));
    k2=h*(x2(t)+h+y2(t)+k1);
    y2(t+1)=y2(t)+k1/2+k2/2;
    x2(t+1)=x2(1)+t*h;
end
y2(k)
!**********************************************************************************!
!third order
x3(1)=x1(1);
y3(1)=y1(1);
for t=1:1:k
    k1=h*(x3(t)+y3(t));
    k2=h*(x3(t)+(h/2)+y3(t)+(k1/2));
    k3=h*(x3(t)+h+y3(t)+2*k2-k1);
    y3(t+1)=y3(t)+((1/6)*(k1+4*k2+k3));
    x3(t+1)=x3(1)+t*h;
end
y3(k)
!***********************************************************************************!
!fourth order
x4(1)=x1(1);
y4(1)=y1(1);
for t=1:1:k
    k1=h*(x4(t));
    k2=h*(x4(t)+h/2);
    k3=h*(x4(t)+h/2);
    k4=h*(x4(t)+h);
    y4(t+1)=y4(t)+(1/6)*(k1+2*k2+2*k3+k4);
    x4(t+1)=x4(1)+t*h;
end
y4(k)