(1) % MotionGenesis file: MGSolveNonlinearEqnContinuously.txt
   (2) % Copyright (c) 2024 Motion Genesis LLC.
   (3) % Purpose: Continuous solution to the nonlinear algebraic equation:
   (4) %          y^4 - 8*y - 0.3*t^2 - 9*sin(t) = 0
   (5) %---------------------------------------------------------------------------
   (6) Variable  y'
   (7) eqnToSolve = y^4 - 8*y - 0.3*t^2 - 9*sin(t)
-> (8) eqnToSolve = y^4 - 0.3*t^2 - 9*sin(t) - 8*y

   (9) yValuesAt0 = GetQuarticRoots( Evaluate(eqnToSolve, t=0) = 0,  y )
-> (10) yValuesAt0 = [-1 - 1.732051*imaginary;  -1 + 1.732051*imaginary;  -1.721328E-18;  2]

   (11) %---------------------------------------------------------------------------
   (12) %   Differentiate eqnToSolve to form an equation that is _linear_ in y'.
   (13) eqnDt = Dt( eqnToSolve )
-> (14) eqnDt = 4*y^3*y' - 0.6*t - 9*cos(t) - 8*y'

   (15) %---------------------------------------------------------------------------
   (16) %   Solve eqnDt = 0 to form an ODE (ordinary differential equation) for y'.
   (17) Solve( eqnDt = 0,   y' )
-> (18) y' = 0.15*(t+15*cos(t))/(-2+y^3)

   (19) %---------------------------------------------------------------------------
   (20) %   Set the initial value for y from one of the quadratic roots at t = 0.
   (21) Input  y = yValuesAt0[4] meters
   (22) %---------------------------------------------------------------------------
   (23) %   List output quantities (for subsequent ODE command).
   (24) Output t seconds,  y meters
   (25) %---------------------------------------------------------------------------
   (26) %   Set numerical integration parameters, solve ODEs, and plot results.
   (27) Input  tFinal = 20 seconds,  tStep = 0.02 sec,  absError = 1.0E-07
   (28) ODE() MGSolveNonlinearEqnContinuously

   (29) Plot  MGSolveNonlinearEqnContinuously.1
   (30) %--------------------------------------------------------------------
   (31) %   Record input together with responses.