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 polynomial_series_sol
 formal power series solutions with polynomial coefficients for a linear ODE

 Calling Sequence polynomial_series_sol(ode, var,opts) polynomial_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 polynomial_series_sol command returns one formal power series solution or a set of formal power series solutions of the given linear ordinary differential equation with polynomial coefficients. The ODE must be either homogeneous or inhomogeneous with a right-hand side that is a polynomial, a rational function, or a "nice" power series (see LODEstruct) in the independent variable $x$.
 • 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 linear in var
 – ode must have polynomial coefficients in $x$
 – 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 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. 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 routine selects such formal power series solutions where $v\left(n\right)$ is a polynomial for all sufficiently large $n$.

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. The default is $a=0$. 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 this option is given, then the command returns one formal power series solution at a with polynomial coefficients if it exists; otherwise, it returns NULL. If a is not given, it returns a set of formal power series solutions with polynomial coefficients for all possible points that are determined by Slode[candidate_points](ode,var,'type'='polynomial').
 • '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.
 • 'index'=n
 Specifies a name for the summation index in the power series. The default value is the global name _n.

Examples

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\left(3{x}^{2}-6x+3\right)\left(\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)+\left(12x-12\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+6y\left(x\right)$
 ${\mathrm{ode}}{:=}\left({3}{}{{x}}^{{2}}{-}{6}{}{x}{+}{3}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}\left({12}{}{x}{-}{12}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{6}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{polynomial_series_sol}\left(\mathrm{ode},y\left(x\right)\right)$
 $\left\{{\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\mathrm{_n}}{}{{\mathrm{_C}}}_{{1}}{+}{{\mathrm{_C}}}_{{0}}\right){}{{x}}^{{\mathrm{_n}}}\right\}$ (2)

Inhomogeneous equations are handled:

 > $\mathrm{polynomial_series_sol}\left(\mathrm{ode}=\frac{6\left(180{x}^{2}-150x+25+3{x}^{4}-42{x}^{3}\right)}{{\left(x-5\right)}^{3}},y\left(x\right),'\mathrm{index}'=n\right)$
 $\left\{{80}{-}{24}{}{x}{-}{25}{}\left({\sum }_{{n}{=}{2}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{\left({x}{-}{4}\right)}^{{n}}\right)\right\}$ (3)