Why should the numerical solution give the same answer as the analytical one? Looking at pixels overlapping on the screen is not a very precise way to discern anything. You should examine the error between the two (absolute and/or relative). You might also want to examine what happens when you change the step size. And you might want to play with a linear system as well. You don't need to integrate out so far to see these effects – just setting t
equal 0.1
or 1
suffices. Here is some better-formatted code to work with:
t0 = 0;
dt = 0.01;
tf = 0.1;
t = t0:dt:tf; % No need to integrate t in for loop for fixed time step
lt = length(t);
f = zeros(1,lt); % Pre-allocate f
f0 = 1000; % Initial condition
f(1) = f0;
a = 10;
for i = 1:lt-1
f(i+1) = f(i) + a*f(i)*dt;
%f(i+1) = f(i) + a*dt; % Alternative linear system to try
end
% Analytic solution
fa = f0*exp(a*t);
%fa = f0+a*t; % Alternative linear system to try
figure;
plot(t,f,'r-',t,fa,'bo')
% Plot absolute error
figure;
plot(t,abs(f-fa))
% Plot relative error
figure;
plot(t,abs(f-fa)./fa)
You're also not preallocating any of your arrays which makes you code very inefficient. My code does. Read about that here.
Much more beyond this is really off-topic for this site, which is focussed on programming rather than mathematics. If you really have questions about the numerical details that aren't answered by reading your text book (or the Wikipedia page for the Euler method) then you should ask them at Math.StackExchange.