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Slode

 rational_series_sol
 formal power series solutions with rational coefficients for a linear ODE

 Calling Sequence rational_series_sol(ode, var,opts) rational_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 rational_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 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 rational function 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. 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 rational coefficients if it exists; otherwise, it returns NULL. If a is not given, it returns a set of formal power series solutions with rational coefficients for all possible points that are determined by Slode[candidate_points](ode,var,'type'='rational').
 • '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.
 • 'index'=n
 Specifies a name for the summation index in the power series. The default value is the global name _n.

Examples

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode1}≔2x\left(x-1\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)+\left(7x-3\right)\mathrm{diff}\left(y\left(x\right),x\right)+2y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}{2}{}{x}{}\left({x}{-}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({7}{}{x}{-}{3}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{rational_series_sol}\left(\mathrm{ode1},y\left(x\right),x=0\right)$
 ${2}{}{{\mathrm{_C}}}_{{1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({\mathrm{_n}}{+}{1}\right){}{{x}}^{{\mathrm{_n}}}}{{2}{}{\mathrm{_n}}{+}{1}}\right)$ (2)
 > $\mathrm{ode2}≔\left(3-x\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)-\mathrm{diff}\left(y\left(x\right),x\right)$
 ${\mathrm{ode2}}{≔}\left({3}{-}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (3)
 > $\mathrm{rational_series_sol}\left(\mathrm{ode2},y\left(x\right),'\mathrm{index}'=n\right)$
 $\left\{{{\mathrm{_C}}}_{{1}}{+}{{\mathrm{_C}}}_{{0}}{}\left({\sum }_{{n}{=}{1}}^{{\mathrm{\infty }}}{}\frac{{\left({x}{-}{2}\right)}^{{n}}}{{n}}\right)\right\}$ (4)

An inhomogeneous equation:

 > $\mathrm{ode3}≔-2y\left(x\right)+\left(-2x+2{x}^{2}\right)\mathrm{diff}\left(y\left(x\right),x,x,x\right)+\left(13x-2{x}^{2}-5\right)\mathrm{diff}\left(y\left(x\right),x,x\right)+\left(12-7x\right)\mathrm{diff}\left(y\left(x\right),x\right)=\frac{13}{6{x}^{3}}+\left(\mathrm{Sum}\left(\frac{{x}^{-n}\left(-12+13{n}^{2}+4{n}^{4}-17{n}^{3}+14n\right)}{\left(n-2\right)\left(n-3\right)\left(n-1\right)n},n=4..\mathrm{\infty }\right)\right)$
 ${\mathrm{ode3}}{≔}{-}{2}{}{y}{}\left({x}\right){+}\left({2}{}{{x}}^{{2}}{-}{2}{}{x}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{2}{}{{x}}^{{2}}{+}{13}{}{x}{-}{5}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({12}{-}{7}{}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}\frac{{13}}{{6}{}{{x}}^{{3}}}{+}\left({\sum }_{{n}{=}{4}}^{{\mathrm{\infty }}}{}\frac{{{x}}^{{-}{n}}{}\left({4}{}{{n}}^{{4}}{-}{17}{}{{n}}^{{3}}{+}{13}{}{{n}}^{{2}}{+}{14}{}{n}{-}{12}\right)}{\left({n}{-}{2}\right){}\left({n}{-}{3}\right){}\left({n}{-}{1}\right){}{n}}\right)$ (5)
 > $\mathrm{rational_series_sol}\left(\mathrm{ode3},y\left(x\right),'\mathrm{free}'=A\right)$
 ${\sum }_{{\mathrm{_n}}{=}{2}}^{{\mathrm{\infty }}}{}\frac{{2}{}{{\mathrm{_n}}}^{{3}}{}{{A}}_{{1}}{-}{4}{}{{\mathrm{_n}}}^{{2}}{}{{A}}_{{1}}{+}{2}{}{\mathrm{_n}}{}{{A}}_{{1}}{+}{2}{}{\mathrm{_n}}{-}{1}}{\left({\mathrm{_n}}{-}{1}\right){}{\mathrm{_n}}{}\left({2}{}{\mathrm{_n}}{-}{1}\right){}{{x}}^{{\mathrm{_n}}}}$ (6)