Why would you use an ODE solver to determine equilibrium points for an ODE? Yes, I know it somehow seems logical. But this is not what ODE solvers are designed to do.
An equilibrium point is a point where the function does not change. Just set the derivatives to zero, then solve. That is, an equilibrium point is a point where Y' = 0. The same applies where you have a system of equations. As such, you use fsolve or solve or vpasolve to find that point or points, NOT an ODE solver.
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