Ordinary Differential Equations.

@article{Parke1958OrdinaryDE,
  title={Ordinary Differential Equations.},
  author={Nathan Grier Parke and Wilfred Kaplan},
  journal={American Mathematical Monthly},
  year={1958},
  volume={67},
  pages={96}
}
together with the initial condition y(t0) = y0 A numerical solution to this problem generates a sequence of values for the independent variable, t0, t1, . . . , and a corresponding sequence of values for the dependent variable, y0, y1, . . . , so that each yn approximates the solution at tn yn ≈ y(tn), n = 0, 1, . . . Modern numerical methods automatically determine the step sizes hn = tn+1 − tn so that the estimated error in the numerical solution is controlled by a specified tolerance. The… 

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References

9660e-11 2
  • 9660e-11 2