Runge-Kutta with a function that changes at each time step

I previously asked a question about solving an equation dx/dt = u(x,t) where u(x,t) is a velocity. I want to solve for the positions x to obtain the trajectory x(t). Here, I have a starting position x0 and compute the velocity at that initial point, u(x0,t0). I want to use Runge-Kutta to determine the time evolution of the position based on the velocitie. I can calculate an arbitrary instantaneous velocity u(x,t) from the forces acting on a position x. I think I'm close, but need to modify the following to take into account the arguments I'm passing to the velocity function, and then calculate the time evolution of all points / particles (two in this case). How do I do that? I'm not sure how to pass the arguments into the velocity function for one thing.

Here's the simplified current version:

% Input
f = [0.05;0;0]; % Force
N = 2; % Number of points
% Initial positions
x1 = [-0.5,0,0];
x2 = [0.5,0,0];
deltaX = x2-x1;
c = 3; % Constant
x = [x1;x2];
% Define second force
fs = @(deltaX) c*deltaX;
% Initial values
t0 = 0;
tf = 2.5;
sol = ode45(@(t,x) u(x,t),[t0 tf],[x1(1) x2(1)]);

function v = u(x,t)
% Calculate forces on each point
fs_x1 = fs(deltaX);
fs_x2 = -fs_x1;
% Calculate net forces
f_k1 = -f + fs_x1(:);
f_k2 = f + fs_x2(:);
f_k = [f_k1(1);f_k1(2);f_k1(3);f_k2(1);f_k2(2);f_k2(3)];
% Calculate velocities
v = S*f_k; % I have a separate function for the calculation of S, which is a function of the positions
end

[Sam Chak]

Suggest you show math model of the dynamic system instead of the code so that we can understand the problem clearly.

Also show how the matrix S is calculated.

[Sam Chak]

@L'O.G., have you shown the full equations of motion? It looks a 6th-order system.

Answer 1

[Bruno Luong]

Again everything is written in the doc of ode45 "For information on how to provide additional parameters to the function odefun, see Parameterizing Functions."

Just a question of reading.