Slode
candidate_mpoints
determine mpoints for msparse power series solutions
Calling Sequence
Parameters
Description
Examples
candidate_mpoints(ode, var)
candidate_mpoints(LODEstr)
ode

homogeneous linear ODE with polynomial coefficients
var
dependent variable, for example y(x)
LODEstr
LODEstruct data structure
The candidate_mpoints command determines for all positive integers $m$ candidate points for msparse power series solutions of the given homogeneous linear ordinary differential equation with polynomial coefficients, called mpoints.
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.
This command returns a list of lists with three elements:
an integer ${m}_{i}\>1$, the sparse order;
a LODEstruct representing an ${m}_{i}$sparse differential equation with constant coefficients which is a right factor of the given equation;
a set of candidate ${m}_{i}$points.
The list is sorted by sparse order.
If for some sparseorder $m$ the given equation has a nontrivial msparse right factor with constant coefficients, then the equation has msparse power series solutions at an arbitrary point, and these solutions are solutions of this right factor. If the set of candidate mpoints is not empty, then the equation may or may not have msparse power series solutions at such a point, but it does not have msparse power series solutions at any point outside this set.
$\mathrm{with}\left(\mathrm{Slode}\right)\:$
$\mathrm{ode}\u2254\left(2+{x}^{2}\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)x+\left(2+{x}^{2}\right)\mathrm{diff}\left(y\left(x\right)\,x\right)2xy\left(x\right)$
${\mathrm{ode}}{\u2254}\left({{x}}^{{2}}{+}{2}\right){}\left(\frac{{{\ⅆ}}^{{3}}}{{\ⅆ}{{x}}^{{3}}}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{2}{}\left(\frac{{{\ⅆ}}^{{2}}}{{\ⅆ}{{x}}^{{2}}}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{+}\left({{x}}^{{2}}{+}{2}\right){}\left(\frac{{\ⅆ}}{{\ⅆ}{x}}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{2}{}{x}{}{y}{}\left({x}\right)$
$\mathrm{candidate\_mpoints}\left(\mathrm{ode}\,y\left(x\right)\right)$
$\left[\left[{2}{\,}{\mathrm{LODEstruct}}{}\left(\left\{{y}{}\left({x}\right){+}\frac{{{\ⅆ}}^{{2}}}{{\ⅆ}{{x}}^{{2}}}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right\}{\,}\left\{{y}{}\left({x}\right)\right\}\right){\,}\left\{{0}\right\}\right]\right]$
See Also
LODEstruct
Slode[candidate_points]
Slode[msparse_series_sol]
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