function Example3
% clc
% clear
close all

global a;

% change constant 'a' and see the problems with numerical solution due to the computer arithmetic representation of number...
a=1;

y0=[1;1]; %[x;y]
tmax=2*pi;

% try different ode solvers - ode45, ode23, ode15s
odeSolve=ode45(@f_dydt,[0,tmax],y0); %[uc, i]

%%%%%%%%% analytic solution %%%%%%%%%
dt=1000;
t=[0:tmax/dt:tmax];
x=exp(a*sin(t));
y=exp(-a*sin(t));
%%%%%%%%%%%%%%%%%%


figure
plot(odeSolve.x, odeSolve.y,'o',t,x,t,y)
grid on
title('Example 3')
legend('numerical solution', 'analytical solution')

fprintf('Number of integration steps: %g\n',length(odeSolve.y));

x_anal=exp(a*sin(odeSolve.x));
y_anal=exp(-a*sin(odeSolve.x));
fprintf('Error ||v_num-v_anal||: %g\n',norm(odeSolve.y-[x_anal;y_anal]));
end


function dydt = f_dydt(t,y)
global a
dydt = zeros(2,1);    % a column vector
dydt(1) =  a*y(1)*cos(t);
dydt(2) =  -a*y(2)*cos(t);
end