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Slode

 dAlembertian_series_sol
 formal power series solutions with d'Alembertian coefficients for a linear ODE

 Calling Sequence dAlembertian_series_sol(ode,var,opts) dAlembertian_series_sol(LODEstr,opts)

Parameters

 ode - linear ODE with polynomial coefficients var - dependent variable, for example y(x) opts - optional arguments of the form keyword=value LODEstr - LODEstruct data structure

Description

 • The dAlembertian_series_sol command returns one formal power series solution or a set of formal power series solutions with d'Alembertian coefficients for the given linear ordinary differential equation with polynomial coefficients.
 • If ode is an expression, then it is equated to zero.
 • The command returns an error message if the differential equation ode does not satisfy the following conditions.
 – ode must be linear in var
 – ode must be homogeneous or have a right-hand side that is rational or a "nice" power series in $x$
 – The coefficients of ode must be polynomial in the independent variable of var, for example, $x$, over the rational number field which can be extended by one or more parameters.
 • A homogeneous linear ordinary differential equation with coefficients that are polynomials in $x$ has a linear space of formal power series solutions ${\sum }_{n=0}^{\mathrm{\infty }}v\left(n\right){P}_{n}\left(x\right)$ where ${P}_{n}\left(x\right)$ is one of ${\left(x-a\right)}^{n}$, $\frac{{\left(x-a\right)}^{n}}{n!}$, $\frac{1}{{x}^{n}}$, or $\frac{1}{{x}^{n}n!}$, $a$ is the expansion point, and the sequence $v\left(n\right)$ satisfies a homogeneous linear recurrence. In the case of an inhomogeneous equation with a right-hand side that is a "nice" power series, $v\left(n\right)$ satisfies an inhomogeneous linear recurrence.
 • The command selects such formal power series solutions where $v\left(n\right)$ is a d'Alembertian sequence, that is, $v\left(n\right)$ is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).
 • The command determines an integer $N\ge 0$ such that $v\left(n\right)$ can be represented in the form of a d'Alembertian term:

$v\left(n\right)={h}_{1}\left(n\right)\left({\sum }_{{n}_{1}=N}^{n-1}{h}_{2}\left({n}_{1}\right)\left({\sum }_{{n}_{2}=N}^{{n}_{1}-1}\mathrm{...}\left({\sum }_{{n}_{s}=N}^{{n}_{s-1}-1}{h}_{s+1}\left({n}_{s}\right)\right)\right)\right)\mathrm{\left( + \right)}$

 for all $n\ge N$, where ${h}_{i}\left(n\right)$, $1\le i\le s+1$, is a hypergeometric term (see SumTools[Hypergeometric]):

${h}_{i}\left(n\right)={h}_{i}\left(N\right)\left({\prod }_{k=N}^{n-1}R\left(k\right)\right)\mathrm{\left( ++ \right)}$

 such that $R\left(k\right)=\frac{{h}_{i}\left(k+1\right)}{{h}_{i}\left(k\right)}$ is rational in $k$ for all $k\ge N$.

Options

 • x=a or 'point'=a
 Specifies the expansion point in the case of a homogeneous equation or an inhomogeneous equation with rational right-hand side. It can be an algebraic number, depending rationally on some parameters, or $\mathrm{\infty }$. In the case of a "nice" series right-hand side the expansion point is given by the right-hand side and cannot be changed. If the point is given, then the command returns one formal power series solution at a with d'Alembertian coefficients if it exists; otherwise, it returns NULL. If $a$  is not given, it returns a set of formal power series solutions with d'Alembertian coefficients for all singular points of ode as well as one generic ordinary point.
 • 'free'=C
 Specifies a base name C to use for free variables C, C, etc. The default is the global name  _C. Note that the number of free variables may be less than the order of the given equation.
 • 'indices'=[n,k]
 Specifies base names for dummy variables. The default values are the global names _n and _k, respectively. The name n is used as the summation index in the power series. the names n1, n2, etc., are used as summation indices in ( + ). The name k is used as the product index in ( ++ ).
 • 'outputHGT'=name
 Specifies the form of representation of hypergeometric terms.  The default value is 'inert'.
 – 'inert' - the hypergeometric term ( ++ ) is represented by an inert product, except for ${\prod }_{k=N}^{n-1}1$, which is simplified to $1$.
 – 'rcf1' or 'rcf2' - the hypergeometric term is represented in the first or second minimal representation, respectively (see ConjugateRTerm).
 – 'active' - the hypergeometric term is represented by non-inert products which, if possible, are computed (see product).
 • 'outputDAT'=name
 Specifies the form of representation of the sums in ( + ). The default is 'inert'.
 – 'inert' - the sums are in the inert form, except for trivial sums of the form ${\sum }_{k=u}^{v-1}1$, which are simplified to $v-u$.
 – 'gosper' - Gosper's algorithm (see Gosper) is used to find a closed form for the sums in ( + ), if possible, starting with the innermost one.

Examples

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\left({x}^{2}+x-2\right)\mathrm{diff}\left(y\left(x\right),x,x\right)+\left({x}^{2}-x\right)\mathrm{diff}\left(y\left(x\right),x\right)-\left(6{x}^{2}+7x\right)y\left(x\right)$
 ${\mathrm{ode}}{≔}\left({{x}}^{{2}}{+}{x}{-}{2}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({{x}}^{{2}}{-}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\left({6}{}{{x}}^{{2}}{+}{7}{}{x}\right){}{y}{}\left({x}\right)$ (1)
 > $\mathrm{dAlembertian_series_sol}\left(\mathrm{ode},y\left(x\right),'\mathrm{outputHGT}'='\mathrm{active}','\mathrm{indices}'=\left[n,k\right]\right)$
 $\left\{\frac{{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{n}{=}{1}}^{{\mathrm{\infty }}}{}\frac{{{2}}^{{n}}{}{\left({x}{-}{1}\right)}^{{n}}}{{\mathrm{\Gamma }}{}\left({n}\right)}\right)}{{2}}{,}\left(\frac{{315}{}\left({\sum }_{{n}{=}{7}}^{{\mathrm{\infty }}}{}\frac{{{2}}^{{n}}{}\left({n}{-}{6}\right){}{\left({x}{+}{2}\right)}^{{n}}}{{\mathrm{\Gamma }}{}\left({n}{+}{1}\right)}\right)}{{8}}{-}\frac{{4095}}{{4}}{-}\frac{{1575}{}{x}}{{4}}{-}{315}{}{\left({x}{+}{2}\right)}^{{2}}{-}\frac{{315}{}{\left({x}{+}{2}\right)}^{{3}}}{{2}}{-}\frac{{105}{}{\left({x}{+}{2}\right)}^{{4}}}{{2}}{-}\frac{{21}{}{\left({x}{+}{2}\right)}^{{5}}}{{2}}\right){}{{\mathrm{_C}}}_{{0}}{+}\left({-}{17143}{}\left({\sum }_{{n}{=}{7}}^{{\mathrm{\infty }}}{}\frac{{{2}}^{{n}}{}\left({n}{-}{6}\right){}\left({\sum }_{{\mathrm{n1}}{=}{7}}^{{n}{-}{1}}{}\frac{{\left(\frac{{1}}{{6}}\right)}^{{\mathrm{n1}}}{}{\mathrm{\Gamma }}{}\left({\mathrm{n1}}{+}{1}\right)}{\left({\mathrm{n1}}{-}{5}\right){}\left({\mathrm{n1}}{-}{6}\right)}\right){}{\left({x}{+}{2}\right)}^{{n}}}{{\mathrm{\Gamma }}{}\left({n}{+}{1}\right)}\right){-}\frac{{40}{}\left({\sum }_{{n}{=}{7}}^{{\mathrm{\infty }}}{}\frac{{{2}}^{{n}}{}\left({n}{-}{6}\right){}\left({\sum }_{{\mathrm{n1}}{=}{7}}^{{n}{-}{1}}{}\frac{{\left(\frac{{1}}{{6}}\right)}^{{\mathrm{n1}}}{}{\mathrm{\Gamma }}{}\left({\mathrm{n1}}{+}{1}\right){}\left({\sum }_{{\mathrm{n2}}{=}{7}}^{{\mathrm{n1}}{-}{1}}{}\frac{\left({\mathrm{n2}}{-}{5}\right){}{\left({-9}\right)}^{{\mathrm{n2}}}}{{\mathrm{\Gamma }}{}\left({\mathrm{n2}}\right)}\right)}{\left({\mathrm{n1}}{-}{5}\right){}\left({\mathrm{n1}}{-}{6}\right)}\right){}{\left({x}{+}{2}\right)}^{{n}}}{{\mathrm{\Gamma }}{}\left({n}{+}{1}\right)}\right)}{{9}}{+}\frac{{997295}}{{486}}{+}\frac{{383575}{}{x}}{{486}}{+}\frac{{153430}{}{\left({x}{+}{2}\right)}^{{2}}}{{243}}{+}\frac{{77435}{}{\left({x}{+}{2}\right)}^{{3}}}{{243}}{+}\frac{{73295}{}{\left({x}{+}{2}\right)}^{{4}}}{{729}}{+}\frac{{18835}{}{\left({x}{+}{2}\right)}^{{5}}}{{729}}{-}\frac{{2540}{}{\left({x}{+}{2}\right)}^{{6}}}{{729}}\right){}{{\mathrm{_C}}}_{{1}}{,}\left(\left({\sum }_{{\mathrm{_n}}{=}{9}}^{{\mathrm{\infty }}}{}\frac{\left({\mathrm{_n}}{-}{8}\right){}{{2}}^{{\mathrm{_n}}}{}{\left({x}{+}{3}\right)}^{{\mathrm{_n}}}}{{512}{}{\mathrm{_n}}{!}}\right){-}\frac{{25}}{{256}}{-}\frac{{7}{}{x}}{{256}}{-}\frac{{3}{}{\left({x}{+}{3}\right)}^{{2}}}{{128}}{-}\frac{{5}{}{\left({x}{+}{3}\right)}^{{3}}}{{384}}{-}\frac{{\left({x}{+}{3}\right)}^{{4}}}{{192}}{-}\frac{{\left({x}{+}{3}\right)}^{{5}}}{{640}}{-}\frac{{\left({x}{+}{3}\right)}^{{6}}}{{2880}}{-}\frac{{\left({x}{+}{3}\right)}^{{7}}}{{20160}}\right){}{{\mathrm{_C}}}_{{0}}{+}\left({-}\frac{{528299}{}\left({\sum }_{{\mathrm{_n}}{=}{9}}^{{\mathrm{\infty }}}{}\frac{{1024}{}\left({\mathrm{_n}}{-}{8}\right){}{{2}}^{{\mathrm{_n}}}{}\left({\sum }_{{\mathrm{_n1}}{=}{9}}^{{\mathrm{_n}}{-}{1}}{}\frac{{\left(\frac{{1}}{{8}}\right)}^{{\mathrm{_n1}}}{}{\mathrm{\Gamma }}{}\left({\mathrm{_n1}}{+}{1}\right)}{\left({\mathrm{_n1}}{-}{7}\right){}\left({\mathrm{_n1}}{-}{8}\right)}\right){}{\left({x}{+}{3}\right)}^{{\mathrm{_n}}}}{{2835}{}{\mathrm{_n}}{!}}\right)}{{2198016}}{+}\left({\sum }_{{\mathrm{_n}}{=}{9}}^{{\mathrm{\infty }}}{}{-}\frac{{2}{}\left({\mathrm{_n}}{-}{8}\right){}{{2}}^{{\mathrm{_n}}}{}\left({\sum }_{{\mathrm{_n1}}{=}{9}}^{{\mathrm{_n}}{-}{1}}{}\frac{{\left(\frac{{1}}{{8}}\right)}^{{\mathrm{_n1}}}{}{\mathrm{\Gamma }}{}\left({\mathrm{_n1}}{+}{1}\right){}\left({\sum }_{{\mathrm{_n2}}{=}{9}}^{{\mathrm{_n1}}{-}{1}}{}\frac{\left({\mathrm{_n2}}{-}{7}\right){}{\left({-12}\right)}^{{\mathrm{_n2}}}{}\left({I}{}\sqrt{{11}}{-}{2}{}{\mathrm{_n2}}{-}{7}\right){}\left({I}{}\sqrt{{11}}{+}{2}{}{\mathrm{_n2}}{+}{7}\right)}{\left({\mathrm{_n2}}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({\mathrm{_n2}}{+}{1}\right)}\right)}{\left({\mathrm{_n1}}{-}{7}\right){}\left({\mathrm{_n1}}{-}{8}\right)}\right){}{\left({x}{+}{3}\right)}^{{\mathrm{_n}}}}{{19683}{}\left({I}{}\sqrt{{11}}{-}{25}\right){}\left({I}{}\sqrt{{11}}{+}{25}\right){}{\mathrm{_n}}{!}}\right){+}\frac{{869574607}}{{273468358656}}{+}\frac{{243522833}{}{x}}{{273468358656}}{+}\frac{{3853999}{}{\left({x}{+}{3}\right)}^{{2}}}{{5064228864}}{+}\frac{{366367}{}{\left({x}{+}{3}\right)}^{{3}}}{{859963392}}{+}\frac{{1269989}{}{\left({x}{+}{3}\right)}^{{4}}}{{7596343296}}{+}\frac{{11983373}{}{\left({x}{+}{3}\right)}^{{5}}}{{227890298880}}{+}\frac{{1166693}{}{\left({x}{+}{3}\right)}^{{6}}}{{113945149440}}{+}\frac{{572503}{}{\left({x}{+}{3}\right)}^{{7}}}{{265872015360}}{-}\frac{{93781}{}{\left({x}{+}{3}\right)}^{{8}}}{{398808023040}}\right){}{{\mathrm{_C}}}_{{1}}\right\}$ (2)
 > $\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right),x\right)-y\left(x\right)=\mathrm{Sum}\left(\frac{\left(\mathrm{Sum}\left(\mathrm{\Gamma }\left(2\mathrm{n1}+1\right),\mathrm{n1}=0..n-1\right)\right){x}^{n}}{n!},n=0..\mathrm{\infty }\right)$
 ${\mathrm{ode2}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{y}{}\left({x}\right){=}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({\sum }_{{\mathrm{n1}}{=}{0}}^{{n}{-}{1}}{}{\mathrm{\Gamma }}{}\left({2}{}{\mathrm{n1}}{+}{1}\right)\right){}{{x}}^{{n}}}{{n}{!}}$ (3)
 > $\mathrm{dAlembertian_series_sol}\left(\mathrm{ode2},y\left(x\right),'\mathrm{outputHGT}'='\mathrm{active}','\mathrm{indices}'=\left[n,k\right]\right)$
 $\left({\sum }_{{n}{=}{1}}^{{\mathrm{\infty }}}{}\frac{\left({n}{-}{1}{+}\left({\sum }_{{\mathrm{n1}}{=}{1}}^{{n}{-}{1}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{n2}}{=}{1}}^{{\mathrm{n1}}{-}{1}}{}{\mathrm{\Gamma }}{}\left({2}{}{\mathrm{n2}}{+}{1}\right)\right)\right){}{{x}}^{{n}}}{{n}{!}}\right){+}{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{{x}}^{{n}}}{{n}{!}}\right)$ (4)

Compatibility

 • The Slode[dAlembertian_series_sol] command was updated in Maple 2017.
 • The ode parameter was updated in Maple 2017.