FPseries - Maple Help

Slode

 FPseries
 formal power series solutions for a linear ODE
 FTseries
 formal Taylor series solutions for a linear ODE

 Calling Sequence FPseries(ode, var, vn, opts) FTseries(ode, var, vn, opts) FPseries(LODEstr, vn, opts) FTseries(LODEstr, vn, opts)

Parameters

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

Description

 • The FPseries and FTseries commands return a power series solution of the given linear ordinary differential equation with polynomial coefficients which must be 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 commands choose a polynomial basis depending on the expansion point. If $a$ is a rational or algebraic number, then the basis is ${\left(x-a\right)}^{n}$ for FPseries and $\frac{{\left(x-a\right)}^{n}}{n!}$ for FTseries. If $a=\mathrm{\infty }$, then the basis is $\frac{1}{{x}^{n}}$ for FPseries and $\frac{1}{{x}^{n}n!}$ for FTseries.
 • Both commands then build a linear recurrence $Rv\left(n\right)=0$ for the sequence of coefficients of the series solution in the chosen basis. They determine the number $M$ of initial values of this sequence so that it is possible to calculate any element from these initial values using the recurrence (taking into account the order and the zeros of the leading coefficient of the recurrence), and calculate these initial values.
 • The formal power series is represented by an FPSstruct data structure:

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

 where $v\left(0\right),\mathrm{...},v\left(M\right)$ are expressions, the initial series coefficients.
 • In the case of an inhomogeneous equation with a rational right hand side the commands work similarly. In the case of a "nice" power series right hand side the recurrence $Rv\left(n\right)$ is inhomogeneous.

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. Otherwise, the default is $a=0$.
 • 'terms'=N
 Specifies a nonnegative integer N, the required number of initial terms of the series solutions. If this option is given, then the command compares it with the determined number $M$ and, if $M, calculates the first N initial coefficients.
 • '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

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\left(3{x}^{2}-6x+3\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)+\left(12x-12\right)\mathrm{diff}\left(y\left(x\right),x\right)+6y\left(x\right)$
 ${\mathrm{ode}}{≔}\left({3}{}{{x}}^{{2}}{-}{6}{}{x}{+}{3}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({12}{}{x}{-}{12}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{6}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{FPseries}\left(\mathrm{ode},y\left(x\right),v\left(n\right)\right)$
 ${\mathrm{FPSstruct}}{}\left({{\mathrm{_C}}}_{{1}}{}{x}{+}{{\mathrm{_C}}}_{{0}}{+}\left({\sum }_{{n}{=}{2}}^{{\mathrm{\infty }}}{}{v}{}\left({n}\right){}{{x}}^{{n}}\right){,}\left({{n}}^{{2}}{-}{n}\right){}{v}{}\left({n}\right){+}\left({-}{2}{}{{n}}^{{2}}{+}{2}{}{n}\right){}{v}{}\left({n}{-}{1}\right){+}\left({{n}}^{{2}}{-}{n}\right){}{v}{}\left({n}{-}{2}\right)\right)$ (2)
 > $\mathrm{FPseries}\left(\mathrm{ode},y\left(x\right),v\left(n\right),x=1\right)$
 ${\mathrm{FPSstruct}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{v}{}\left({n}\right){}{\left({x}{-}{1}\right)}^{{n}}{,}\left({{n}}^{{2}}{+}{3}{}{n}{+}{2}\right){}{v}{}\left({n}\right)\right)$ (3)
 > $\mathrm{FTseries}\left(\mathrm{ode},y\left(x\right),v\left(n\right),'\mathrm{free}'=A,'\mathrm{terms}'=2\right)$
 ${\mathrm{FPSstruct}}{}\left({{A}}_{{0}}{+}{{A}}_{{1}}{}{x}{+}\frac{\left({-}{2}{}{{A}}_{{0}}{+}{4}{}{{A}}_{{1}}\right){}{{x}}^{{2}}}{{2}}{+}\left({\sum }_{{n}{=}{3}}^{{\mathrm{\infty }}}{}\frac{{v}{}\left({n}\right){}{{x}}^{{n}}}{{n}{!}}\right){,}{v}{}\left({n}\right){-}{2}{}{n}{}{v}{}\left({n}{-}{1}\right){+}\left({{n}}^{{2}}{-}{n}\right){}{v}{}\left({n}{-}{2}\right)\right)$ (4)

Some inhomogeneous examples:

 > $\mathrm{FPseries}\left(\mathrm{ode}={\left(x-1\right)}^{4},y\left(x\right),v\left(n\right),x=1\right)$
 ${\mathrm{FPSstruct}}{}\left(\frac{{\left({x}{-}{1}\right)}^{{4}}}{{90}}{+}\left({\sum }_{{n}{=}{5}}^{{\mathrm{\infty }}}{}{v}{}\left({n}\right){}{\left({x}{-}{1}\right)}^{{n}}\right){,}\left({3}{}{{n}}^{{2}}{+}{9}{}{n}{+}{6}\right){}{v}{}\left({n}\right)\right)$ (5)
 > $\mathrm{FPseries}\left(\mathrm{ode}=\mathrm{Sum}\left(\frac{1}{{x}^{n}},n=3..\mathrm{\infty }\right),y\left(x\right),v\left(n\right)\right)$
 ${\mathrm{FPSstruct}}{}\left(\frac{{{\mathrm{_C}}}_{{0}}}{{x}}{+}\frac{{{\mathrm{_C}}}_{{1}}}{{{x}}^{{2}}}{+}\left({\sum }_{{n}{=}{3}}^{{\mathrm{\infty }}}{}\frac{{v}{}\left({n}\right)}{{{x}}^{{n}}}\right){,}\left({3}{}{{n}}^{{2}}{-}{9}{}{n}{+}{6}\right){}{v}{}\left({n}\right){+}\left({3}{}{{n}}^{{2}}{-}{9}{}{n}{+}{6}\right){}{v}{}\left({n}{-}{2}\right){+}\left({-}{6}{}{{n}}^{{2}}{+}{18}{}{n}{-}{12}\right){}{v}{}\left({n}{-}{1}\right){-}{1}\right)$ (6)