function example1_rk(lambda,N)

% EXAMPLE1_RK solve ODE with Runge Kutta method
%   example1_rk(lambda,N) computes a sequence of time points t
%   (in the interval [0,1]) and approximate solutions U1 and U2 at those time 
%   points to the problem u'(t)=lambda*u with u(0)=1 (with exact solution
%   u(t)=exp(lambda*t)).
%   The approximate solution U1 is computed using the explicit Euler method
%   with N+1 timesteps and the approximate solution U2 uses the improved
%   Euler scheme

% exact solution
uexact=@(t,lambda) exp(lambda*t);
figure(1), clf
fplot(@(t) uexact(t,lambda),[0,1],'b','linewidth',2), hold on

t=linspace(0,1,N+1);
dt=t(2)-t(1);

% vector for explicit Euler solutions
U1=zeros(1,N+1);
% initial condition
U1(1)=1;

% vector for improved Euler solutions
U2=U1;

for n=1:N
    U1(n+1)=U1(n)*(1+lambda*dt);
    U2(n+1)=U2(n)*(1+lambda*dt+(lambda*dt)^2/2);
end

figure(1)
plot(t,U1,'r',t,U2,'g','linewidth',2)
xlabel('t','fontsize',20), ylabel('u','fontsize',20)
legend('exact solution','explicit Euler','improved Euler','fontsize',20,'location','NorthWest')
title(['Solution with N=',num2str(N),' and lambda=',num2str(lambda)],'fontsize',20)
% exact solution at grid points
Ue=uexact(t,lambda);
figure(2),clf
plot(t,(U1-Ue),'r',t,(U2-Ue),'g','linewidth',2)
xlabel('t','fontsize',20), ylabel('error','fontsize',20)
legend('error in explicit Euler','error in improved Euler','location','SouthWest','fontsize',20)
title(['Error with N=',num2str(N),' and lambda=',num2str(lambda)],'fontsize',20)

disp(['Error in explicit Euler solution at t=1 is ',num2str(abs(U1(end)-Ue(end)))])
disp(['Error in improved Euler solution at t=1 is ',num2str(abs(U2(end)-Ue(end)))])