Differential equation system solving

Hi everyone, can you help me please with this?

I would like to solve a system of differential equations for several conditions (like a for loop of solutions, where the initial conditions of the n+1 equations system are the solutions founded on by the solution of the n system, additionally for each time (t) that I solve the system I will need to change the constants of the equations (which are time dependant).

I would really appreciate any help you can provide.