STABILITY of Solution of ODE (and a bit of stability of Method of finding solution of ODE)

To show that is the analytical solution of the problem, we need to show

Stability of ODE solution is determined by the perturbations of solution that do not diverge away over time. Based on the graph, we could conclude that the solution is asymptotically stable for and stable for . In other word, if the solution stable, the solution is convergent as .

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S(x)011.22541.32491.38101.4170

Taylor series suggest that if however, would as well approach the actual solution if when for any. Hence we could conclude from our result that perturbation would result in significant effect to the solution as