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 msparse_series_sol
 formal m-sparse power series solutions for a linear ODE

 Calling Sequence msparse_series_sol(ode, var, vn, opts) msparse_series_sol(LODEstr, vn, opts)

Parameters

 ode - linear ODE with polynomial coefficients var - dependent variable, for example y(x) vn - new function in the form $v\left(n\right)$ opts - optional arguments of the form keyword=value LODEstr - LODEstruct data structure

Description

 • The msparse_series_sol command returns a set of m-sparse power series solutions of the given linear ordinary differential equation with polynomial coefficients.
 • If ode is an expression, then it is equated to zero.
 • The routine returns an error message if the differential equation ode does not satisfy the following conditions.
 – ode must be homogeneous and linear in var
 – ode must have polynomial coefficients in the independent variable of var, for example, $x$
 – The coefficients of ode must be either rational numbers or depend rationally on 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.
 • This routine selects such formal power series solutions where for an integer $m\ge 2$ there is an integer $i$ such that
 – $v\left(n\right)\ne 0$ only if $n-i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}m=0$, and
 – $v\left(\left(n+1\right)m+i\right)=p\left(n\right)v\left(mn+i\right)$ for all sufficiently large $n$, where $p\left(n\right)$ is a rational function.
 • The m-sparse power series is represented by an FPSstruct data-structure (see Slode[FPseries]):

$\mathrm{FPSstruct}\left(v\left(0\right)+v\left(1\right){P}_{1}\left(x\right)+\mathrm{...}+v\left(M\right){P}_{M}\left(x\right)+{\sum }_{n=M+1}^{\infty }v\left(mn+N\right){P}_{mn+N}\left(x\right),\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}Rv\left(mn+N\right)\right);$

 where
 – $v\left(0\right)$,...,$v\left(M\right)$ are expressions, the initial series coefficients,
 – $M$ is a nonnegative integer, and
 – $s$ is an integer such that $M+1\le ms+N$.

Options

 • x=a or 'point'=a
 Specifies the expansion point a. The default is $a=0$. It can be an algebraic number, depending rationally on some parameters, or $\mathrm{\infty }$.
 If this option is given, then the command returns a set of m-sparse power series solutions at the given point a. Otherwise, it returns a set of m-sparse power series solutions for all possible points that are determined by Slode[candidate_mpoints](ode,var).
 • 'sparseorder'=m0
 Specifies an integer m0. If this option is given, then the procedure computes a set of m-sparse power series solutions with $m=\mathrm{m0}$ only. Otherwise, it returns a set of m-sparse power series solution for all possible values of $m$.
 If both an expansion point and a sparse order are given, then the command can also compute a set of m-sparse series solutions for an inhomogeneous equation with polynomial coefficients and a right-hand side that is rational in the independent variable $x$. Otherwise, the equation has to be homogeneous.
 • '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 if the expansion point is singular.

Examples

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\frac{2\left({x}^{3}-\frac{2{x}^{2}}{3}+\frac{1x}{9}\right)\left(\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)\right)}{9}+\frac{2\left(9{x}^{2}-4x+\frac{1}{3}\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)}{9}+\frac{2\left(18x-4\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)}{9}+\frac{4y\left(x\right)}{3}$
 ${\mathrm{ode}}{≔}\frac{{2}}{{9}}{}\left({{x}}^{{3}}{-}\frac{{2}}{{3}}{}{{x}}^{{2}}{+}\frac{{1}}{{9}}{}{x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{y}{}\left({x}\right)\right){+}\frac{{2}}{{9}}{}\left({9}{}{{x}}^{{2}}{-}{4}{}{x}{+}\frac{{1}}{{3}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}\frac{{2}}{{9}}{}\left({18}{}{x}{-}{4}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}\frac{{4}}{{3}}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{msparse_series_sol}\left(\mathrm{ode},y\left(x\right),v\left(n\right)\right)$
 $\left\{{\mathrm{FPSstruct}}{}\left({{\mathrm{_C}}}_{{0}}{+}{\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{v}{}\left({2}{}{n}\right){}{\left({x}{-}\frac{{1}}{{6}}\right)}^{{2}{}{n}}{,}{-}{36}{}{v}{}\left({2}{}{n}{-}{2}\right){+}{v}{}\left({2}{}{n}\right)\right){,}{\mathrm{FPSstruct}}{}\left({{\mathrm{_C}}}_{{1}}{}\left({x}{-}\frac{{1}}{{6}}\right){+}{\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{v}{}\left({2}{}{n}{+}{1}\right){}{\left({x}{-}\frac{{1}}{{6}}\right)}^{{2}{}{n}{+}{1}}{,}{-}{36}{}{v}{}\left({2}{}{n}{-}{1}\right){+}{v}{}\left({2}{}{n}{+}{1}\right)\right)\right\}$ (2)

Inhomogeneous equations are handled:

 > $\mathrm{ode1}≔{z}^{2}\left(\frac{{ⅆ}^{2}}{ⅆ{z}^{2}}y\left(z\right)\right)+3z\left(\frac{ⅆ}{ⅆz}y\left(z\right)\right)+\left({z}^{2}+1-{n}^{2}\right)y\left(z\right)=1$
 ${\mathrm{ode1}}{≔}{{z}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}{}{y}{}\left({z}\right)\right){+}{3}{}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}{}{y}{}\left({z}\right)\right){+}\left({-}{{n}}^{{2}}{+}{{z}}^{{2}}{+}{1}\right){}{y}{}\left({z}\right){=}{1}$ (3)
 > $\mathrm{msparse_series_sol}\left(\mathrm{ode1},y\left(z\right),v\left(k\right),z=\mathrm{∞},'\mathrm{sparseorder}'=2\right)$
 $\left\{{\mathrm{FPSstruct}}{}\left(\frac{{1}}{{{z}}^{{2}}}{+}{\sum }_{{k}{=}{2}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{v}{}\left({2}{}{k}\right)}{{{z}}^{{2}{}{k}}}{,}{v}{}\left({2}{}{k}\right){+}\left({4}{}{{k}}^{{2}}{-}{{n}}^{{2}}{-}{12}{}{k}{+}{9}\right){}{v}{}\left({2}{}{k}{-}{2}\right)\right)\right\}$ (4)