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DEtools

 regularsp
 compute the regular singular points of a second order non-autonomous linear ODE

 Calling Sequence regularsp(des, ivar, dvar)

Parameters

 des - second order linear ordinary differential equation or its list form ivar - indicates the independent variable when des is a list with the ODE coefficients dvar - indicates the dependent variable, required only when des is an ODE and the dependent variable is not obvious

Description

 • 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

 p(x) y''(x) + q(x) y'(x) + r(x) y(x) = 0,  p(x) <> 0,  p'(x) <> 0

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

 1) alpha is a singular point, 2) limit( (x-alpha)*q(x)/p(x), x=alpha ) = 0 and limit( (x-alpha)^2*r(x)/p(x), x=alpha ) = 0.

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

Examples

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

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

An ordinary differential equation (ODE)

 > $\mathrm{ODE}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)=\left(\frac{\mathrm{α}}{x-1}+\frac{\mathrm{β}}{x}+\frac{\mathrm{γ}}{{x}^{2}}+\frac{\mathrm{δ}}{{\left(x-1\right)}^{2}}+{\mathrm{λ}}^{2}\right)y\left(x\right)$
 ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}\left(\frac{{\mathrm{α}}}{{x}{-}{1}}{+}\frac{{\mathrm{β}}}{{x}}{+}\frac{{\mathrm{γ}}}{{{x}}^{{2}}}{+}\frac{{\mathrm{δ}}}{{\left({x}{-}{1}\right)}^{{2}}}{+}{{\mathrm{λ}}}^{{2}}\right){}{y}{}\left({x}\right)$ (1)
 > $\mathrm{regularsp}\left(\mathrm{ODE}\right)$
 Warning, DEtools[regularsp] has been superseded by DEtools[singularities]
 $\left[{0}{,}{1}\right]$ (2)
 > $\mathrm{singularities}\left(\mathrm{ODE}\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}\right\}{,}{\mathrm{irregular}}{=}\left\{{\mathrm{∞}}\right\}$ (3)

The coefficient list form

 > $\mathrm{coefs}≔\left[21\left({x}^{2}-x+1\right),0,100{x}^{2}{\left(x-1\right)}^{2}\right]:$
 > $\mathrm{regularsp}\left(\mathrm{coefs},x\right)$
 $\left[{0}{,}{1}{,}{\mathrm{∞}}\right]$ (4)
 > $\mathrm{singularities}\left(\mathrm{coefs},x\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}{,}{\mathrm{∞}}\right\}{,}{\mathrm{irregular}}{=}\left\{{}\right\}$ (5)

You can convert convert an ODE to the coefficient list form using DEtools[convertAlg] form

 > $\mathrm{ODE}≔\left(2{x}^{2}+5{x}^{3}\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+\left(5x-{x}^{2}\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\left(\frac{1}{x}+x\right)y\left(x\right)=0$
 ${\mathrm{ODE}}{≔}\left({5}{}{{x}}^{{3}}{+}{2}{}{{x}}^{{2}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}\left({-}{{x}}^{{2}}{+}{5}{}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}\left(\frac{{1}}{{x}}{+}{x}\right){}{y}{}\left({x}\right){=}{0}$ (6)
 > $L≔\mathrm{convertAlg}\left(\mathrm{ODE},y\left(x\right)\right)$
 ${L}{≔}\left[\left[\frac{{1}}{{x}}{+}{x}{,}{-}{{x}}^{{2}}{+}{5}{}{x}{,}{5}{}{{x}}^{{3}}{+}{2}{}{{x}}^{{2}}\right]{,}{0}\right]$ (7)
 > $\mathrm{regularsp}\left(L,x\right)$
 $\left[{-}\frac{{2}}{{5}}{,}{\mathrm{∞}}\right]$ (8)
 > $\mathrm{singularities}\left(L,x\right)$
 ${\mathrm{regular}}{=}\left\{{-}\frac{{2}}{{5}}{,}{\mathrm{∞}}\right\}{,}{\mathrm{irregular}}{=}\left\{{0}\right\}$ (9)