clear
format compact

%%
% Consider the ODEs $u'=u$ and $u'=-u$. In each case we compute $\partial
% f/\partial u = \pm 1$, so the condition number bound is $e^{(b-a)}$ in both
% problems. However, they behave quite differently. In the case of
% exponential growth, $u'=u$, the bound is the actual
% condition number.
for u0 = [0.7 1 1.3]
    fplot(@(t) exp(t)*u0,[0 3])
    hold on
end

xlabel('t')
ylabel('u(t)')
title('Exponential divergence of solutions')
grid on

%%
% But with $u'=-u$, solutions actually get closer together with time.
clf
for u0 = [0.7 1 1.3]
    fplot(@(t) exp(-t)*u0,[0 3])
    hold on
end
xlabel('t')
ylabel('u(t)')
title('Exponential convergence of solutions')
grid on

%%
% In this case the actual condition number is one, due to the difference of
% solutions at the initial time.