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INTRODUCTION: The free-fall trajectories of objects (geodesics) near a massive black hole are treated using Einstein's “general theory of relativity” equations (GR) [1] and are contrasted to corresponding solutions using “Newton’s Universal Law of Gravitation” equations (NULG). The effect of Einstein’s S=1/(r-Rs) ODE singularity coefficient on trajectories approaching the black hole (as spherical coordinate r→Rs ) is investigated, where Rs is the Schwarzchild radius 2GM/c. Assistance with numerical integration as S goes from -∞ to +∞ as r crosses r=Rs is treated with a smooth cutoff transition function S(r)n (Figure 4 inset) with cutoff off values 1.2x10n { n=4,2,-2,-6}. The paper illustrates solving for the position vector solution x⃗(λ) as a solution to Einstein's time dependent GR nonlinear ODE’s equations (GR) where λ is a proper time parameter for advancing the position . For comparative purposes, it is of interest to examine how GR compares to the same solution using Newton’s (NULG) equations. USE OF COMSOL MULTIPHYSICS: Nonstandard ODE’s often encountered in physics research is a good fit for applying COMSOL’s general equation solving capabilities via the Equation Based Modeling Global-ODE and DAE Interface study. RESULTS: A series of increasing complex models are examined . Four example illustrations are addressed: 1) satellite crashes earth 2) Moon orbits earth (Fig.1), 3) “plunge fall” towards massive black hole starting at r=Ro=1.5Rs and 4) “death spiral fall” towards black hole starting at larger r=Ro=3Rs . In situations where we are far from the r→Rs region , we see that GR and NULG essentially overlay each other such as in examples 1 and 2 ( S(r)n not in play here). Figure(1) also illustrates good solution comparisons between COMSOL and Mathematica) and predicted the correct orbit period for a set of observed initial conditions. However in examples 3 and 4, the mass of the primary body is the size of the KY Cygni star whose mass is 100 times the mass of our sun. The star is assumed (for exploratory purposes) to be collapsed into a point mass and motions experienced by a secondary body approaching with large tangential component approach velocity is investigated. in the region of r→Rs Here substantial differences between the two theories are experienced, such as the example illustrated in Figures 2-4. CONCLUSION: A sequence of increasing complex examples has shown the importance of the S=1/(r-Rs) singularity coefficient over a range of { n=4,2,-2,-6} cutoff values. Everyday models like satellite tracking and moon orbit earth trajectories, illustrate GR and NULG give essentially the same results, however when approaching a massive black hole as r→Rs, the differences are substantial, and solution sensitivity to the singularity transition function cutoff value is significant.