Specifying tolerances for ode45
Example problem
clearvars; clf
Consider the initial value problem
Here the exact solution is 
. We can easily verify this by plugging this into the ODE.
. We can easily verify this by plugging this into the ODE.We plot the direction field and the exact solution together:
f = @(t,y) 2*y-sin(t)-2*cos(t);
dirfield(f,0:.25:6,-1.5:.25:1.5); hold on
T = 6;
fplot(@(t)cos(t),[0 T],'k') % plot cos(t) as black curve
title('exact solution of IVP')
Solve with ode45 using default tolerances
We use ode45 to solve this IVP for 
[ts,ys] = ode45(f,[0,T],1);
plot(ts,ys,'b')
title('solution obtained by ode45 (blue)')
At 
we should get for 
the value
we should get for the valuecos(T)
But the solution from ode45 gives
ys(end)
Solve with ode45 using smaller tolerances
We now choose smaller values for RelTol (default 1e-3) and AbsTol (default 1e-6):
opt = odeset('RelTol',1e-5,'AbsTol',1e-8); % try using even smaller values
[ts,ys] = ode45(f,[0,T],1,opt); % solve with options given by opt
plot(ts,ys,'g')
title('solution obtained by ode45 with smaller tolerances (green)')
hold off
At 
we should get for the value
we should get for the valuecos(T)
Now the solution from ode45 gives
ys(end)
Try to use even smaller values for the tolerances!