Solutions of differential and difference equations

$\Sigma{}^{{\rm it}}$ provides efficient solvers for linear ordinary differential and difference equations and systems. All the solvers are usually called kernel, the exact functionality depends on the type used. The following tables indicate which solver to use depending on the type of equation and the desired type of solutions.

Table: $\Sigma{}^{{\rm it}}$ solvers for polynomial solutions

Equation type	Polynomial solver
Differential equations	LinearOrdinaryOperatorPolynomialSolutions
Recurrence equations	LinearOrdinaryOperatorPolynomialSolutions
Differential systems	LinearOrdinaryFirstOrderSystemPolynomialSolutions
Recurrence systems	LinearOrdinaryFirstOrderSystemPolynomialSolutions

Note that the same types are used to compute the polynomial solutions of differential and difference equations and systems. However, the $RXY$ and $FX$ parameters to the polynomial solvers tell them which type of equation to solve. For differential equations and systems, $RXY$ should be a type of category LinearOrdinaryDifferentialOperatorCategory and $FX$ should be a type of category UnivariatePolynomialCategory. For recurrence equations and systems, $RXY$ should be a type of category LinearOrdinaryRecurrenceCategory and $FX$ should be the type UnivariateFactorialPolynomial since the solutions are computed as factorial polynomials.

Table: $\Sigma{}^{{\rm it}}$ solvers for rational solutions

Equation type	Rational solver
Differential equations	N/A
Recurrence equations	LinearOrdinaryRecurrenceRationalSolutions
Differential systems	N/A
Recurrence systems	LinearOrdinaryRecurrenceSystemRationalSolutions

Table: $\Sigma{}^{{\rm it}}$ solvers for series solutions

Equation type	Series solver
Differential equations	LinearOrdinaryDifferentialOperatorSeriesSolutions
Recurrence equations	N/A
Differential systems	LinearOrdinaryDifferentialSystemSeriesSolutions
Recurrence systems	N/A

Note that the series solver are meant for solving only around an ordinary point of the equation or system.

In addition, LinearOrdinaryRecurrenceHypergeometricSolutions provides a hypergeometric solver for linear recurrences. When solving systems, you must first create the system as an element of LinearOrdinaryFirstOrderSystem. That type does not specify whether its elements represent differential or difference equations, they are interpreted differently by the various solvers depending on the other parameters.