This is an elementary example to illustrate how one might use Chebfun to solve an ODE initial-value problem. We take the world's second-simplest such problem,
$$ u''+ u = 0 , ~~~~ u(0) = 1, ~ u'(0) = 0 $$
on the interval $[0,100]$. The solution is $\cos(x)$.
d = [0,100]; % domain x = chebfun('x',d); % x variable L = chebop(d); % name of operator L.op = @(u) diff(u,2) + u; % linear operator defining the ODE L.lbc = @(u) [u-1;diff(u)]; % imposing Dirichlet and Neumann BCs u = L\0; % solve the problem plot(u,'linewidth',1.6) % plot the solution err = norm(u-cos(x),inf); % measure the error FS = 'fontsize'; xlabel('x',FS,12) ylabel('cos(x)',FS,12) title(sprintf('Solution of IVP for cos(x) -- error = %7.2e',err),FS,14) ylim([-2 2])