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In this paper, we
propose and analyze some schemes of the integral collocation formulation based
on Legendre polynomials. We implement these formulae to solve numerically
Riccati, Logistic and delay differential equations with variable coefficients.
The properties of the Legendre polynomials are used to reduce the proposed
problems to the solution of non-linear system of algebraic equations using
Newton iteration method. We give numerical results to satisfy the accuracy and
the applicability of the proposed schemes.