Let L be a linear differential operator, given as a polynomial in Dx with univariate polynomial coefficients in x over a field $k$ of characteristic zero. The command Desingularize(L,Dx,x) constructs a linear differential operator R such that any solution of $L\left(y\right)=0$ is also a solution of $R\left(y\right)=0$ and R has no apparent singularities.  The operator R is said to maximally desingularize L, and will be right divisible by L over the field $k\left(x\right)$.

An apparent singularity is a point $p$ where the leading coefficient of L vanishes, yet $p$ is not a pole of any holomorphic solution of $L\left(y\right)=0$. In this case there will exist $d$ linearly independent solutions at $p$ where $d$ is the order of L.

A function may be specified using the optional argument func. It is applied to the coefficients of the collected result. Often simplify or factor will be used.

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

$L≔\left(24x^3-18x^4+x^8+6x^5-x^6\right){\mathrm{Dx}}^7+\left(6x^5+72x^3-30x^4-8x^7-72x^2\right){\mathrm{Dx}}^6+\left(-144x^2+36x^6+72x^3-2x^7+144x-18x^4\right){\mathrm{Dx}}^5+\left(24x^3+36x^6+144x-144-72x^2-120x^5-8x^7-x^{10}+x^8\right){\mathrm{Dx}}^4+\left(-24x^5-x^{10}-6x^7+x^8+18x^6\right){\mathrm{Dx}}^3+\left(36x^5-6x^6-72x^4+2x^9\right){\mathrm{Dx}}^2+\left(-36x^4+12x^5-10x^8+2x^9\right)\mathrm{Dx}+64x^7-12x^4-32x^8+8x^9+x^{12}-x^{10}$

$L≔\left(x^8-x^6+6x^5-18x^4+24x^3\right){\mathrm{Dx}}^7+\left(-8x^7+6x^5-30x^4+72x^3-72x^2\right){\mathrm{Dx}}^6+\left(-2x^7+36x^6-18x^4+72x^3-144x^2+144x\right){\mathrm{Dx}}^5+\left(-x^{10}+x^8-8x^7+36x^6-120x^5+24x^3-72x^2+144x-144\right){\mathrm{Dx}}^4+\left(-x^{10}+x^8-6x^7+18x^6-24x^5\right){\mathrm{Dx}}^3+\left(2x^9-6x^6+36x^5-72x^4\right){\mathrm{Dx}}^2+\left(2x^9-10x^8+12x^5-36x^4\right)\mathrm{Dx}+64x^7-12x^4-32x^8+8x^9+x^{12}-x^{10}$

$M≔\mathrm{Desingularize}\left(L\,\mathrm{Dx}\,x\,\mathrm{factor}\right)$

$M≔1728252{\mathrm{Dx}}^8+\left(54154x^7+161694x^6+263753x^5+452649x^4-324882x^3+1728252\right){\mathrm{Dx}}^7+\left(-433232x^6-1293552x^5-2218332x^4-2969808x^3+974646x^2+1728252\right){\mathrm{Dx}}^6+\left(-108308x^6+1626156x^5+5185170x^4+9891042x^3+9684162x^2-1949292x+1728252\right){\mathrm{Dx}}^5+\left(-54154x^9-161694x^8-263753x^7-560957x^6+976266x^5-2924106x^4-10379136x^3-27461604x^2-45113328x+3677544\right){\mathrm{Dx}}^4-x\left(54154x^8+161694x^7+263753x^6+452649x^5-324882x^4+1728252x+13826016\right){\mathrm{Dx}}^3+\left(108308x^8+323388x^7+635814x^6+253914x^5+974646x^4-1728252x^2-10369512x-20739024\right){\mathrm{Dx}}^2+\left(108308x^8-218152x^7-981126x^6-2600232x^5-520824x^4-1728252x^2-6913008x-10369512\right)\mathrm{Dx}+54154x^{11}+161694x^{10}+263753x^9+560957x^8-759650x^7+538258x^6+2595900x^5+10387932x^4+31684620x^3-1728252x^2-3456504x-3456504$

$Q,R≔\mathrm{op}\left(\mathrm{DEtools}\left['\mathrm{rightdivision}'\right]\left(M\,L\,\left[\mathrm{Dx}\,x\right]\right)\right)\:$

$Q$

$\frac{1728252\mathrm{Dx}}{x^3\left(x^5-x^3+6x^2-18x+24\right)}+\frac{54154x^7+161694x^6+263753x^5+452649x^4-324882x^3+1728252}{x^3\left(x^5-x^3+6x^2-18x+24\right)}$

$R$

$0$

Tsai, H. "Weyl closure of a linear differential operator." Journal of Symbolic Computation Vol. 29 No. 4-5 (2000): 747-775.

Chyzak, F.; Dumas, P.; Le, H.Q.; Martins, J.; Mishna, M.; Salvy, B. "Taming apparent singularities via Ore closure." In preparation.

The DEtools[Desingularize] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.