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.

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

$\mathrm{ode}≔\left(2+x^2\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)-2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)x+\left(2+x^2\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-2xy\left(x\right)$

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

$\mathrm{candidate\_mpoints}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[\left[2\,\mathrm{LODEstruct}\left(\left\{y\left(x\right)+\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right\}\,\left\{y\left(x\right)\right\}\right)\,\left\{0\right\}\right]\right]$