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 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 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 command 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 $n-i\mathbf{mod}m=0$, and

the sequence $v\left(\mathrm{mn}+i\right)$ satisfies a linear recurrence $\mathrm{Rv}\left(\mathrm{mn}+i\right)=0$ for all sufficiently large $n$.

The m-sparse power series is represented by an FPSstruct data-structure (see Slode[FPseries]):

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 $M+1\le ms+N$.

x=a or 'point'=a

Specifies the expansion point a. 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 command 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.

$\mathrm{with}\left(\mathrm{Slode}\right)\:$

$\mathrm{ode}≔\frac{2}{9}\left(x^3-\frac{2}{3}{x}^2+\frac{1}{9}x\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+\frac{2}{9}\left(9x^2-4x+\frac{1}{3}\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\frac{2}{9}\left(18x-4\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+\frac{4}{3}y\left(x\right)$

$\mathrm{ode}≔\frac{2\left(x^3-\frac{2}{3}{x}^2+\frac{1}{9}x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)}{9}+\frac{2\left(9x^2-4x+\frac{1}{3}\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{9}+\frac{2\left(18x-4\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{9}+\frac{4y\left(x\right)}{3}$

$\mathrm{msparse\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\,v\left(n\right)\right)$

$\left\{\mathrm{FPSstruct}\left({\mathrm{\_C}}_0+\left(\sum _{n=1}^{\mathrm{\infty }}v\left(2n\right){\left(x-\frac{1}{6}\right)}^{2n}\right)\,-36v\left(2n-2\right)+v\left(2n\right)\right)\,\mathrm{FPSstruct}\left({\mathrm{\_C}}_1\left(x-\frac{1}{6}\right)+\left(\sum _{n=1}^{\mathrm{\infty }}v\left(2n+1\right){\left(x-\frac{1}{6}\right)}^{2n+1}\right)\,-36v\left(2n-1\right)+v\left(2n+1\right)\right)\right\}$

$\mathrm{ode1}≔z^2\mathrm{diff}\left(y\left(z\right)\,z\,z\right)+3z\mathrm{diff}\left(y\left(z\right)\,z\right)+\left(z^2+1-n^2\right)y\left(z\right)=1$

$\mathrm{ode1}≔z^2\left(\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)\right)+3z\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)+\left(-n^2+z^2+1\right)y\left(z\right)=1$

$\mathrm{msparse\_series\_sol}\left(\mathrm{ode1}\,y\left(z\right)\,v\left(k\right)\,z=\mathrm{\infty }\,'\mathrm{sparseorder}'=2\right)$

$\left\{\mathrm{FPSstruct}\left(\frac{1}{z^2}+\left(\sum _{k=2}^{\mathrm{\infty }}\frac{v\left(2k\right)}{z^{2k}}\right)\,v\left(2k\right)+\left(4k^2-n^2-12k+9\right)v\left(2k-2\right)\right)\right\}$