candidate_mpoints - Maple Help

Slode

 candidate_mpoints
 determine m-points for m-sparse power series solutions

 Calling Sequence candidate_mpoints(ode, var) candidate_mpoints(LODEstr)

Parameters

 ode - homogeneous linear ODE with polynomial coefficients var - dependent variable, for example y(x) LODEstr - LODEstruct data structure

Description

 • The candidate_mpoints command determines for all positive integers $m$ candidate points for m-sparse power series solutions of the given homogeneous linear ordinary differential equation with polynomial coefficients, called m-points.
 • 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 sparse-order $m$ the given equation has a nontrivial m-sparse right factor with constant coefficients, then the equation has m-sparse power series solutions at an arbitrary point, and these solutions are solutions of this right factor. If the set of candidate m-points is not empty, then the equation may or may not have m-sparse power series solutions at such a point, but it does not have m-sparse power series solutions at any point outside this set.

Examples

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\left(2+{x}^{2}\right)\mathrm{diff}\left(y\left(x\right),x,x,x\right)-2\mathrm{diff}\left(y\left(x\right),x,x\right)x+\left(2+{x}^{2}\right)\mathrm{diff}\left(y\left(x\right),x\right)-2xy\left(x\right)$
 ${\mathrm{ode}}{≔}\left({{x}}^{{2}}{+}{2}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{2}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{+}\left({{x}}^{{2}}{+}{2}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{2}{}{x}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{candidate_mpoints}\left(\mathrm{ode},y\left(x\right)\right)$
 $\left[\left[{2}{,}{\mathrm{LODEstruct}}{}\left(\left\{{y}{}\left({x}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right\}{,}\left\{{y}{}\left({x}\right)\right\}\right){,}\left\{{0}\right\}\right]\right]$ (2)