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LinearOperators

 IntegrateSols
 check for the existence of a primitive element, and perform accurate integration

 Calling Sequence IntegrateSols(L, x, case)

Parameters

 L - an Ore operator x - the name of the independent variable case - a parameter indicating the case of the equation ('differential' or 'shift')

Description

 • The LinearOperators[IntegrateSols] function performs "accurate integration". That is, it solves the following problem. Let y satisfy L(y)=0 and g satisfy delta(g)=y, where delta means the usual derivative in the differential case and the first difference in the shift case. The routine builds an annihilator S for g of the same degree as that of L, and an operator K such that g=K(y) if both exist. Otherwise, it returns NULL.
 • An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator $\frac{2}{x}+x\mathrm{D}+\left(x+1\right){\mathrm{D}}^{2}+{\mathrm{D}}^{3}$.
 • There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

Examples

 > $\mathrm{with}\left(\mathrm{LinearOperators}\right):$
 > $\mathrm{expr}≔\sqrt{x}{\mathrm{log}\left(x\right)}^{2}$
 ${\mathrm{expr}}{:=}\sqrt{{x}}{}{{\mathrm{ln}}{}\left({x}\right)}^{{2}}$ (1)

An annihilator for expr is

 > $L≔\mathrm{FactoredAnnihilator}\left(\mathrm{expr},x,'\mathrm{differential}'\right)$
 ${L}{:=}{\mathrm{FactoredOrePoly}}{}\left(\left[\frac{{3}}{{2}{}{x}}{,}{1}\right]{,}\left[\frac{{1}}{{2}{}{x}}{,}{1}\right]{,}\left[{-}\frac{{1}}{{2}{}{x}}{,}{1}\right]\right)$ (2)

which can be written in non-factored form as

 > $L≔\mathrm{FactoredOrePolyToOrePoly}\left(L,x,'\mathrm{differential}'\right)$
 ${L}{:=}{\mathrm{OrePoly}}{}\left({-}\frac{{1}}{{8}{}{{x}}^{{3}}}{,}\frac{{1}}{{4}{}{{x}}^{{2}}}{,}\frac{{3}}{{2}{}{x}}{,}{1}\right)$ (3)
 > $\mathrm{IntegrateSols}\left(L,x,'\mathrm{differential}'\right)$
 ${\mathrm{OrePoly}}{}\left({-}\frac{{27}}{{8}{}{{x}}^{{3}}}{,}\frac{{13}}{{4}{}{{x}}^{{2}}}{,}{-}\frac{{3}}{{2}{}{x}}{,}{1}\right){,}{\mathrm{OrePoly}}{}\left(\frac{{26}}{{27}}{}{x}{,}{-}\frac{{4}}{{9}}{}{{x}}^{{2}}{,}\frac{{8}}{{27}}{}{{x}}^{{3}}\right)$ (4)

References

 Abramov, S. A., and van Hoeij, M. "Integration of Solutions of Linear Functional Equations." Integral Transforms and Special Functions. (1999): 3-12.