I have some questions about Mathematica programming and would appreciate if you could help me.

I want to solve a system of ordinary differential equations \ [Mu] ‘[t] and \ [Lambda]’ [t] and each equation contains a large number of terms so it is impractical to write them explicitly. I express these terms as two functions F1 and F2 that depend on two parameters P1 and P2 and \ [Lambda] [t] and \ [Mu] [t].

I have been able to solve this system for a couple of initial conditions \ [Lambda] [0] = ic1 and \ [Mu] [0] = ic2, but I would like to solve my system of equations for a continuum of values \ [Lambda] [0] = {0, …., Pi / 2} and \ [Mu] [0] = {0, …., Infinity} and then get \ [Lambda] [t] and \ [Mu] [t] and use them to perform an integral on \ [Lambda]= \ [Lambda] [0] ={0, …., Pi / 2} and \ [Mu] =\ [Mu] [0] ={0, …., Infinity} that are precisely our initial conditions.

I integrate the product of a function G in the time t (where \ [Lambda] [t] and \ [Mu] [t] are taken into account for a certain initial condition defined by the continuous ranges of the integral) with the same function, but in t = 0 (where the initial conditions are taken into account with the continuous ranges of the integral).

The structure of the program is:

`ode = {\[Mu]'[t] == F1[p1, p2, \[Lambda][t], \[Mu][t]], \[Lambda]'[t] == F2[p1, p2, \[Lambda][t], \[Mu][t]], \[Mu][0] == {0, ...., Pi/2}, \[Lambda][0] == {0, ...., Infinity}}; Sol = NDSolve[ode, {\[Mu], \[Lambda]}, {t, 0, 1},`Method -> "Some method to choose"] \[Mu]1[t_] := Evaluate[\[Mu][t] /. Sol] // First \[Lambda]1[t_] := Evaluate[\[Lambda][t] /. Sol] // First data = ParallelTable[{t, NIntegrate[ G[p1, p2, \[Mu]1[ t] "for the initial condition \[Mu]=\[Mu][0]", \[Lambda]1[ t] "for the initial condition \[Lambda]=\[Lambda][0]"] G[p1, p2, \[Mu] "=\[Mu][0]", \[Lambda] "=\[Lambda][0]"] , {\[Mu] "=`\[Mu][0](initial condition)", 0, Pi/2}, {\[Lambda] "=\[Lambda][0](initial condition)", 0, Infinity}, Method -> {"Some method to choose"}]}, {t, 0, 1}];` `