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

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

Parameters

 ode - homogeneous 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_formal_sol command returns formal solutions with d'Alembertian series coefficients to the given homogeneous 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 homogeneous and linear in var
 – 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 basis of formal solutions (see DEtools[formal_sol]). A formal solution contains a finite number of power series ${\sum }_{n=0}^{\mathrm{\infty }}v\left(n\right){T}^{n}$ where $T$ is a parameter and the sequence $v\left(n\right)$ satisfies a linear recurrence (homogeneous or inhomogeneous).
 • This command selects such formal solutions that contain only series with d'Alembertian coefficients. A sequence is called d'Alembertian if it 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

 • 'parameter'=T
 Specifies the name T that is used to denote $\mathrm{\lambda }{x}^{\frac{1}{r}}$ where $\mathrm{\lambda }$ is a constant and $r$ is called the ramification index. If this option is given, then the command expresses the formal solutions in terms of T and returns a list of lists each of which is of the form [formal solution, relation between T and x]. Otherwise, it returns the formal solutions in terms of ${x}^{\frac{1}{r}}$.
 • x=a or 'point'=a
 Specifies the expansion point a. It can be an algebraic number, depending rationally on some parameters, or $\mathrm{\infty }$.
 The default is $a=0$.
 • 'free'=C
 Specifies a base name C to use for free variables C[0], C[1], 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

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\left(-4-{x}^{2}+2x\right)y\left(x\right)+\left(2x-3{x}^{3}-{x}^{2}\right)\mathrm{diff}\left(y\left(x\right),x\right)+\left({x}^{3}-{x}^{4}\right)\mathrm{diff}\left(y\left(x\right),x,x\right)$
 ${\mathrm{ode}}{≔}\left({-}{{x}}^{{2}}{+}{2}{}{x}{-}{4}\right){}{y}{}\left({x}\right){+}\left({-}{3}{}{{x}}^{{3}}{-}{{x}}^{{2}}{+}{2}{}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{{x}}^{{4}}{+}{{x}}^{{3}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (1)
 > $\mathrm{dAlembertian_formal_sol}\left(\mathrm{ode},y\left(x\right),'\mathrm{outputHGT}'='\mathrm{active}','\mathrm{indices}'=\left[n,k\right]\right)$
 ${{x}}^{{2}}{}\left({-}\frac{\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{{x}}^{{n}}\right)}{{2}}{+}\frac{\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({\sum }_{{\mathrm{n1}}{=}{0}}^{{n}{-}{1}}{}{\left({-}\frac{{1}}{{2}}\right)}^{{\mathrm{n1}}}{}{\mathrm{\Gamma }}{}\left({\mathrm{n1}}{+}{3}\right){}\left({\mathrm{n1}}{+}{2}\right)\right){}{{x}}^{{n}}\right)}{{4}}\right){}{{\mathrm{_C}}}_{{0}}{+}\frac{{{ⅇ}}^{\frac{{2}}{{x}}}{}\left(\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{{x}}^{{n}}\right){-}\frac{{1}}{{3}}\right){}{{\mathrm{_C}}}_{{1}}}{{x}}$ (2)
 > $\mathrm{ode}≔{\left(x-1\right)}^{2}\mathrm{diff}\left(y\left(x\right),x,x,x\right)-\left(x-1\right)\left(x-7\right)\mathrm{diff}\left(y\left(x\right),x,x\right)-2\left(2x-5\right)\mathrm{diff}\left(y\left(x\right),x\right)-2y\left(x\right)$
 ${\mathrm{ode}}{≔}{\left({x}{-}{1}\right)}^{{2}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\left({x}{-}{1}\right){}\left({x}{-}{7}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{2}{}\left({2}{}{x}{-}{5}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{2}{}{y}{}\left({x}\right)$ (3)
 > $\mathrm{dAlembertian_formal_sol}\left(\mathrm{ode},y\left(x\right),x=0,'\mathrm{outputHGT}'='\mathrm{inert}','\mathrm{indices}'=\left[n,k\right]\right)$
 ${{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{{x}}^{{n}}\right){+}{{\mathrm{_C}}}_{{1}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{n}{}{{x}}^{{n}}\right){+}{{\mathrm{_C}}}_{{2}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\left({\sum }_{{\mathrm{n1}}{=}{0}}^{{n}{-}{1}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{n2}}{=}{0}}^{{\mathrm{n1}}{-}{1}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\prod }_{{k}{=}{0}}^{{\mathrm{n2}}{-}{1}}{}\frac{{1}}{{k}{+}{3}}\right){}{{x}}^{{n}}\right)$ (4)