% Uses Euler's method to compute numerical solution to the ode
%  y'=f(t,y)  y(a)=alpha   a<=t<=b.
% Asks user to input a,b,step size h and initial value alpha
% Assumes f is evaluated in a separate function file f.m
% Plots the result with discrete points. 

a=0;
b=2;
h=input('Please enter the step size');
n=ceil((b-a)/h);
alpha1=pi/6;
alpha2=0;
clear t w 
t=a:h:(a+n*h);
w(1,1)=alpha1;
w(2,1)=alpha2;

for i=1:n
    k1=h*F2(t(i),w(:,i));
    k2=h*F2(t(i)+h/2,w(:,i)+(1/2)*k1);
    k3=h*F2(t(i)+h/2,w(:,i)+(1/2)*k2);
    k4=h*F2(t(i+1),w(:,i)+k3); 
  w(:,i+1)=w(:,i)+ (1/6)*(k1+2*k2+2*k3+k4);
end

hold off

plot(t,w(1,:),'*');


%For the example y'=y^2-t^2+1 y(0)=0.5 
% here we now plot the true solution alongside the 
% approximation.
%hold on
%tvals=0:0.01:1;
%plot(tvals,-(2/5)*exp(2*tvals).*cos(tvals)+(1/5)*exp(2*tvals).*sin(tvals));