Keywords: Nonlinear ordinary differential equations Economic Dynamics Optimization Hyperlliptic Integrals Elliptic Integral Nonlinear boundary value problems Elliptic functions Hypergeoemtric functions Optimal Inventory Mathematica

These research activity is characterized by the seek of analytic solutions, without any loss of complexity for the faced models. The famous statement of Godfrey Harold Hardy (1877-1947) "I could never resist an integral" claryfies the approach: integrals, originated from non linear differential equations, are evaluated by means of Special Functions: Gauss Hypegeometric Function, Euler Gamma Functions, Jacobi Elliptic Functions, Legendre Elliptic Integrals, Mathieu Functions, Appell Functions, Lauricella Functions, Lambert W Functions. When necessary, the quadrature is preceded by transformations of variables (Lie Symmetries). All the obtained solutions have always been validated taking advantage of the power of

*Mathematica*. The use of Computer Algebra also has concurred a collaboration in the area of Combinatorical Analysis. Using the calculation of integrals defined by means of special functions some devout formulas for pi have been settled down going to touch topics to the border between Experimental Mathematics, Analytic Number Theory and Special Functions (elliptic, hyperelliptic and hypergeometric).