  There exists a huge number of numerical methods that iteratively construct
approximations to the solution $y(x)$ of an ordinary differential equation
(ODE) $y'(x)=f(x,y)$ starting from an initial value $y_0=y(x_0)$ and using a
finite approximation step $h$ that influences the accuracy of the obtained
approximation. In this paper, a new framework for solving ODEs is presented for
a new kind of a computer -- the Infinity Computer (it has been patented and its
working prototype exists). The new computer is able to work numerically with
finite, infinite, and infinitesimal numbers giving so the possibility to use
different infinitesimals numerically and, in particular, to take advantage of
infinitesimal values of $h$. To show the potential of the new framework a
number of results is established. It is proved that the Infinity Computer is
able to calculate derivatives of the solution $y(x)$ and to reconstruct its
Taylor expansion of a desired order numerically without finding the respective
derivatives analytically (or symbolically) by the successive derivation of the
ODE as it is usually done when the Taylor method is applied. Methods using
approximations of derivatives obtained thanks to infinitesimals are discussed
and a technique for an automatic control of rounding errors is introduced.
Numerical examples are given.
