clear;
clf;

%%%%%%%%%%%%%
% problem 2 %
%%%%%%%%%%%%%

t = 0;
tstop = pi;
y0 = [1 2];

tspan = [t tstop];

[tt,yy] = ode45('func9',tspan,y0);

% plot numerical solution
figure(1);
plot(tt,yy(:,1), 'ro');
hold on;
plot(tt,yy(:,2), 'b*');
grid on;

% exact solution
n = 100;
h = (tstop - t) / n;
tr = t : h : tstop;
x = exp(tr) .* (  cos(2*tr) +    sin(2*tr));
y = exp(tr) .* (2*cos(2*tr) - 2* sin(2*tr));
plot(tr,x,'r');
plot(tr,y,'b');
legend('x_1 numerical','x_2 numerical','x_1 exact','x_2 exact','Location','Northwest');
xlabel('\it{t}');
ylabel('\it{x_1, x_2}');

%%%%%%%%%%%%%
% problem 3 %
%%%%%%%%%%%%%
clear;

t = 0;
tstop = 2*pi;
y0 = [0 1];

tspan = [t tstop];

[tt,yy] = ode45('func9b',tspan,y0);

% plot numerical solution
figure(2);
plot(tt,yy(:,1), 'ro');
hold on;
%plot(tt,yy(:,2), 'b*');
grid on;

% exact solution
n = 100;
h = (tstop - t) / n;
tr = t : h : tstop;
u = sin(tr);
v = cos(tr);
plot(tr,u,'r');
%plot(tr,v,'b');
legend('u numerical','u exact','Location','southwest');
xlabel('\it{x}');
ylabel('\it{u}');