%   Problem 4.1,
%        compares analytic solution, full simulation and simplification
%   Carol Lucas

%%Problem 4_1 in Matlab
A=[-3 1;200 -100]
init=[1;0]
lam=eig(A)
dt0=A*init
M=[1 1 0 0;0 0 1 1; lam.' 0 0; 0 0 lam.']
b=[init;dt0]
as=inv(M)*b
t=[0:.01:4];

x=as(1)*exp(lam(1)*t)+as(2)*exp(lam(2)*t);
y=as(3)*exp(lam(1)*t)+as(4)*exp(lam(2)*t);
figure(1);clf;plot(t,x,t,y);
xlabel('time');title('Problem 4_1');
mess='Analytic Solution'
pause;hold on;

[t,xs]=ode45('P4_1fun',t,init);

plot(t,xs,'o');
mess='Simulation'
pause;

xr=exp(-t);
yr=2*xr;
plot(t,xr,'m+',t,yr,'r+');
mess='Simplification'