Dear Tony

Please change OOP to,

case[nam_] := 
 Module[{begin, end}, 
  nam[set[{be_, en_}]] := {begin, end} = {be, en};
  sol = NDSolve[{y'[x] == Sin[x^2 + 2 x]/(2 Cos[x/2 + 3 x] + 3), 
     y[0] == 0}, y, {x, 0, 500}, MaxSteps -> 10^6, AccuracyGoal -> 30];
  go[] := Table[{x, y[x] /. sol[[1]]}, {x, begin, end, 0.01}]
  ]

object = {Association["name" -> Unique[k], "range" -> {0, 125}, 
    "kernel" -> kernelList[[1]]], 
   Association["name" -> Unique[k], "range" -> {125, 250}, 
    "kernel" -> kernelList[[2]]], 
   Association["name" -> Unique[k], "range" -> {250, 375}, 
    "kernel" -> kernelList[[3]]], 
   Association["name" -> Unique[k], "range" -> {375, 500}, 
    "kernel" -> kernelList[[4]]]};

Map[ParallelEvaluate[case[#name], #kernel] &, object];
Map[ParallelEvaluate[#name[set[#range]], #kernel] &, object];

Then, execute as,

enter codin1 = SessionTime[]
para1 = ParallelEvaluate[go[]];
SessionTime[] - in1e here

ListPlot[para1]

In the Mathematica parallel calculation style, it should be avoided much communication between controller and nodes. From this point of view, NDSolve should be evaluated on each node.

Hi Tony ! Thanks your reply. NDSolve can be used in the OOP as in my report, "Chaos bifurcation of double pendulums calculation with OOP":

https://community.wolfram.com/groups/-/m/t/1109273

In this example, OOP will run on a single core, however, you can run the program on independent core individually. If you can use standard Mathematica, 8 core Mathematica OOPed program will run simultaneously. Regards.