% solution of x'=x using 
% Euler's algorithm (x)
% 2nd order Taylor (y)
% 2nd order RK (z)

function ode

init = .5; % initial condition
h = .5;      % step size
tm = 10;     % time value

x(1) = init;
y(1) = init;
z(1) = init;
t(1) = 0;

for i=1:(tm/h)
    % Euler
    x(i+1) = x(i) + h * x(i);
    % 2nd order Taylor
    y(i+1) = y(i) + h*y(i) + (h^2/2)*y(i);
    % 2nd order RK
    K1 = h * z(i);
    K2 = h * ( z(i) + K1 );
    z(i+1) = z(i) + ( K1 + K2 )/2;
   
    t(i+1) = t(i) + h;   
end 

plot(t,x,t,y,'-.',t,z,'--')
hold on;
fplot('.5*exp(x)',[0,tm],'k');
legend('Euler','2nd order Taylor','2nd order RK','Exact solution')