Integration when some of the values change over time

I have a function of the form where is the position of the k-th object and is the corresponding force. The force has a functional that is also dependent on . I want to integrate this with respect to time and output the value for each time point. I think the values and would therefore change as a function of time. How do I do this which I imagine would need to be done recursively?

[Torsten]

How do you determine xk ? x can be considered as a constant during time integration ?

Answer 1

[Sulaymon Eshkabilov]

Simply write a code, e.g.:

t=linspace(0, 5, 200);
F = 5*cos(2*pi*t);
xk = 2*sin(2*pi*5*t);
x = linspace(0,25);
for ii = 1:numel(x)
y(ii) = sum((F.*(x(ii)-xk))/norm(x(ii)-xk));
end
subplot(211)
plot(xk, F)
xlabel('x_k')
ylabel('F_k')
grid on
subplot(212)
plot(x, y)
xlabel('x')
ylabel('y')
grid on

[L'O.G.]

What about the integration?

[Sulaymon Eshkabilov]

sum() is integrating

[L'O.G.]

y(x) is defined via a sum. As I mentioned in my post, I want to integrate over that. The confusion for me is to do so while updating the values of x_k (and hence y(x) and F_k).