candidate_points - Maple Help

Slode

 candidate_points
 determine points for power series solutions

 Calling Sequence candidate_points(ode, var, 'points_type'=opt) candidate_points(LODEstr, 'points_type'=opt)

Parameters

 ode - linear ODE with polynomial coefficients var - dependent variable, for example y(x) opt - (optional) type of points; one of dAlembertian, hypergeom, rational, polynomial, or all (the default). LODEstr - LODEstruct data structure

Description

 • The candidate_points command determines candidate points for which power series solutions with d'Alembertian, hypergeometric, rational, or polynomial coefficients of the given linear ordinary differential equation exist.
 • 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 linear in var
 – ode must have polynomial coefficients in $x$ over the rational number field which can be extended by one or more parameters.
 – ode must either be homogeneous or have a right hand side that is rational in $x$
 • If opt=all, the output is a list of three elements:
 – a set of hypergeometric points, which may include the symbol 'any_ordinary_point'
 – a set of rational points;
 – a set of polynomial points.
 Otherwise, the output is the set of the required points.
 • Note that the computation of candidate points for power series solutions with d'Alembertian coefficients is currently considerably more expensive computationally than for the other three types of coefficients.

Examples

 > $\mathrm{with}\left(\mathrm{Slode}\right):$
 > $\mathrm{ode}≔\left(3{x}^{2}-6x+3\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)+\left(12x-12\right)\mathrm{diff}\left(y\left(x\right),x\right)+6y\left(x\right)$
 ${\mathrm{ode}}{≔}\left({3}{}{{x}}^{{2}}{-}{6}{}{x}{+}{3}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({12}{}{x}{-}{12}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{6}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{candidate_points}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}'='\mathrm{polynomial}'\right)$
 $\left\{{0}\right\}$ (2)
 > $\mathrm{candidate_points}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}'='\mathrm{rational}'\right)$
 $\left\{{1}\right\}$ (3)
 > $\mathrm{candidate_points}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}'='\mathrm{hypergeometric}'\right)$
 $\left\{{1}{,}{\mathrm{any_ordinary_point}}\right\}$ (4)
 > $\mathrm{candidate_points}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}'='\mathrm{all}'\right)$
 $\left[\left\{{1}{,}{\mathrm{any_ordinary_point}}\right\}{,}\left\{{1}\right\}{,}\left\{{0}\right\}\right]$ (5)
 > $\mathrm{candidate_points}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}'='\mathrm{dAlembertian}'\right)$
 $\left\{{1}{,}{\mathrm{any_ordinary_point}}\right\}$ (6)
 > $\mathrm{ode1}≔60y\left(x\right)+2x\left(x-30\right)\mathrm{diff}\left(y\left(x\right),x\right)-{x}^{2}\left(2x-27\right)\mathrm{diff}\left(y\left(x\right),x,x\right)+{x}^{3}\left(4x-27\right)\mathrm{diff}\left(y\left(x\right),x,x,x\right)=-\frac{2{x}^{2}\left(-5-330x+60{x}^{4}-1137{x}^{2}+32{x}^{3}\right)}{{\left(x-1\right)}^{6}}$
 ${\mathrm{ode1}}{≔}{60}{}{y}{}\left({x}\right){+}{2}{}{x}{}\left({x}{-}{30}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{{x}}^{{2}}{}\left({2}{}{x}{-}{27}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{{x}}^{{3}}{}\left({4}{}{x}{-}{27}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{-}\frac{{2}{}{{x}}^{{2}}{}\left({60}{}{{x}}^{{4}}{+}{32}{}{{x}}^{{3}}{-}{1137}{}{{x}}^{{2}}{-}{330}{}{x}{-}{5}\right)}{{\left({x}{-}{1}\right)}^{{6}}}$ (7)

Inhomogeneous equations are handled:

 > $\mathrm{candidate_points}\left(\mathrm{ode1},y\left(x\right)\right)$
 $\left[\left\{{0}{,}{1}{,}\frac{{27}}{{4}}{,}{\mathrm{any_ordinary_point}}{,}{\mathrm{RootOf}}{}\left({60}{}{{\mathrm{_Z}}}^{{4}}{+}{32}{}{{\mathrm{_Z}}}^{{3}}{-}{1137}{}{{\mathrm{_Z}}}^{{2}}{-}{330}{}{\mathrm{_Z}}{-}{5}\right)\right\}{,}\left\{{-1}{,}{0}{,}{1}{,}\frac{{23}}{{4}}{,}\frac{{27}}{{4}}{,}{\mathrm{RootOf}}{}\left({49}{}{{\mathrm{_Z}}}^{{4}}{-}{287}{}{{\mathrm{_Z}}}^{{3}}{-}{1418}{}{{\mathrm{_Z}}}^{{2}}{-}{714}{}{\mathrm{_Z}}{-}{45}\right){-}{1}{,}{\mathrm{RootOf}}{}\left({49}{}{{\mathrm{_Z}}}^{{4}}{-}{287}{}{{\mathrm{_Z}}}^{{3}}{-}{1418}{}{{\mathrm{_Z}}}^{{2}}{-}{714}{}{\mathrm{_Z}}{-}{45}\right){,}{\mathrm{RootOf}}{}\left({60}{}{{\mathrm{_Z}}}^{{4}}{+}{32}{}{{\mathrm{_Z}}}^{{3}}{-}{1137}{}{{\mathrm{_Z}}}^{{2}}{-}{330}{}{\mathrm{_Z}}{-}{5}\right)\right\}{,}\left\{{-1}{,}{0}{,}\frac{{23}}{{4}}{,}{\mathrm{RootOf}}{}\left({60}{}{{\mathrm{_Z}}}^{{4}}{+}{32}{}{{\mathrm{_Z}}}^{{3}}{-}{1137}{}{{\mathrm{_Z}}}^{{2}}{-}{330}{}{\mathrm{_Z}}{-}{5}\right){-}{1}\right\}\right]$ (8)

An equation which has d'Alembertian series solutions at any ordinary point but doesn't have hypergeometric ones:

 > $\mathrm{ode2}≔\left(x-1\right)\mathrm{diff}\left(y\left(x\right),x\right)-\left(x-2\right)y\left(x\right)$
 ${\mathrm{ode2}}{≔}\left({x}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\left({x}{-}{2}\right){}{y}{}\left({x}\right)$ (9)
 > $\mathrm{candidate_points}\left(\mathrm{ode2},y\left(x\right),'\mathrm{type}'='\mathrm{hypergeometric}'\right)$
 $\left\{{1}\right\}$ (10)
 > $\mathrm{candidate_points}\left(\mathrm{ode2},y\left(x\right),'\mathrm{type}'='\mathrm{dAlembertian}'\right)$
 $\left\{{1}{,}{\mathrm{any_ordinary_point}}\right\}$ (11)