Important: The regularsp command has been deprecated.  Use the superseding command DEtools[singularities], which computes both the regular and irregular singular points, instead.

The regularsp command determines the regular singular points of a given second order linear ordinary differential equation. The ODE could be given as a standard differential equation or as a list with the ODE coefficients (see DEtools[convertAlg]). Given a linear ODE of the form

a point alpha is considered to be a regular singular point if

The results are returned in a list.  In the event that no regular singular points are found, an empty list is returned.

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

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\left(\frac{\mathrm{\alpha }}{x-1}+\frac{\mathrm{\beta }}{x}+\frac{\mathrm{\gamma }}{x^2}+\frac{\mathrm{\delta }}{{\left(x-1\right)}^2}+{\mathrm{\lambda }}^2\right)y\left(x\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\left(\frac{\mathrm{\alpha }}{x-1}+\frac{\mathrm{\beta }}{x}+\frac{\mathrm{\gamma }}{x^2}+\frac{\mathrm{\delta }}{{\left(x-1\right)}^2}+{\mathrm{\lambda }}^2\right)y\left(x\right)$

$\mathrm{regularsp}\left(\mathrm{ODE}\right)$

$\left[0\,1\right]$

$\mathrm{singularities}\left(\mathrm{ODE}\right)$

$\mathrm{regular}=\left\{0\,1\right\},\mathrm{irregular}=\left\{\mathrm{\infty }\right\}$

$\mathrm{coefs}≔\left[21\left(x^2-x+1\right)\,0\,100x^2{\left(x-1\right)}^2\right]\:$

$\mathrm{regularsp}\left(\mathrm{coefs}\,x\right)$

$\left[0\,1\,\mathrm{\infty }\right]$

$\mathrm{singularities}\left(\mathrm{coefs}\,x\right)$

$\mathrm{regular}=\left\{0\,1\,\mathrm{\infty }\right\},\mathrm{irregular}=\varnothing$

$\mathrm{ODE}≔\left(2x^2+5x^3\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\left(5x-x^2\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+\left(\frac{1}{x}+x\right)y\left(x\right)=0$

$\mathrm{ODE}≔\left(5x^3+2x^2\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(-x^2+5x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(\frac{1}{x}+x\right)y\left(x\right)=0$

$L≔\mathrm{convertAlg}\left(\mathrm{ODE}\,y\left(x\right)\right)$

$L≔\left[\left[\frac{1}{x}+x\,-x^2+5x\,5x^3+2x^2\right]\,0\right]$

$\mathrm{regularsp}\left(L\,x\right)$

$\left[-\frac{2}{5}\,\mathrm{\infty }\right]$

$\mathrm{singularities}\left(L\,x\right)$

$\mathrm{regular}=\left\{-\frac{2}{5}\,\mathrm{\infty }\right\},\mathrm{irregular}=\left\{0\right\}$