Description:

This notebook illustrates DifferentialEquations`LdeApprox` package capability of doing polynomial approximation of an LDE solution with non polynomial coefficients. If the given LDE has non-polynomial coefficients one can use the package procedure ToRatCoeffs which gives a rational approximation of the coefficients. This procedure applies RationalInterpolation function from NumericalMath`Approximations` package to each non-polynomial coefficient of the LDE. It should be pointed out that the coefficients can not involve indeterminate variables except independent one. This restriction comes from NumericalMath`Approximations` package as pure numerical one.
First of all we load the package and define an IVP. Then we use ToRatCoeffs and ApproxSol procedures to find 9-th degree polynomial approximation for the IVP solution on interval x = [-1,1]. After that we find exact solution by Mathematica™ function DSolve. Finally we compare exact and approximate results using Mathematica™ function Plot.
 

This loads the package.

This LDE has non-polynomial coefficient.

Trying to find approximate solution(an error occurs as the LDE has non polynomial coefficients).

Using ToRatCoeffs to convert the LDE into one with polynomial coefficients on the given interval.

Trying ApproxSol for the new LDE.

Using Mathematica™ function DSolve to get the exact solution.

Comparing the results.