%% Differential Equations with Maple Chapter 8

%% A case where ode45 does not give a good answer.

%%
% We look at the initial value problem
%%
% 
% $$y' \,\mbox{--}\, y = \mbox{--}3 e^{\mbox{-}2x}, \quad y(0) = 1.$$
% 
% First we calculate the exact solution.  This is a linear
% equation so it is easy to solve.

syms x
y = simplify(dsolve('Dy = y - 3*exp(-2*x), y(0) = 1', 'x'))


%%
% Now we use *ode45* to find a numeric solution.

f = @(x,y) y - 3*exp(-2*x);
[ta,ya] = ode45(f,[0,16],1); 

%%
% We plot the exact solution and approximate one together.
ezplot(y,[0,16]), hold on
plot(ta,ya,'g'), 
title 'Exact and approximate solutions of an unstable equation'
legend({'Exact', 'Approximate'})
hold off

%%
% The solutions are the same until about x=3 then spread apart. The reason
% for this is that the problem is unstable.  When you can't calculate the
% exact solution, you can calculate the numerical solution with greater
% accuracy, then compare.  If the two results are not fairly close, the
% first one is probably not very accurate.  We do this for this example,
% using *odeset* to improve the accuracy.

options = odeset('AbsTol', 1e-10, 'RelTol', 1e-10);
[tb,yb] = ode45(f,[0,16],1,options);
plot(ta,ya,'g'), hold on
plot(tb,yb,'r')
title 'Approximate solutions of an unstable equation'
legend({'1st Approximate','2nd Approximate'},'Position',[0.1572 0.2304 0.235 0.09836])
hold off

%%
% We see the two solutions start to differ between x=3 and x=4.  At x=10
% the difference starts to increase rapidly.  From this, without knowing 
% the exact solution, we could conclude that the first solution is probably
% not very accurate for x>10.  By x=16 the difference is very large. 

%% 
% Here are the graphs of the two approximate solutions and the exact one. 
% The more accurate approximate solution agrees with the exact one on this 
% interval.  

options = odeset('AbsTol', 1e-10, 'RelTol', 1e-10);
[tb,yb] = ode45(f,[0,16],1,options);
plot(ta,ya,'g'), hold on
plot(tb,yb,'r')
ezplot(y,[0,16])
title 'Exact and approximate solutions of an unstable equation'
legend({'1st Approximate','2nd Approximate','Exact'})
hold off