Analytic approximation methods try to compute approximations to the exact solutions on suitable domains, in the form of truncated series expansions over a system of basis functions. In the SciPy stack, we have an implementation based on the Taylor series, through the routine `odefun`

in the module `sympy.mpmath`

.

### Note

`mpmath`

is a Python library for arbitrary-precision floating-point arithmetic, hosted inside the `sympy`

module. Although it is independent of the `numpy`

machinery, they both work well together.

For more information about this library, read the official documentation at http://mpmath.org/doc/current/.

Let's see it in action, first with our trivial example *y'(t) = y(t)*, *y(0) = 1*. The key here is to assess the speed and the accuracy of the approximation, as compared to the actual solution in the interval *[0, 1]*. Its syntax is very simple, we assume the equation is always in the form of *y' = F*, and we provide the routine `odefun`

with this functional *F* and the...