DifferentialAlgebra - Maple Help

Home : Support : Online Help : Mathematics : Differential Equations : Differential Algebra : DifferentialAlgebra/Equations

DifferentialAlgebra

 Equations
 returns the equations of a differential ideal, optionally satisfying given criteria
 Inequations
 returns the inequations of a differential ideal

 Calling Sequence Equations(ideal, criterion, opts) Equations(L, criterion, R, opts) Inequations(ideal, opts) Inequations(L, R, opts)

Parameters

 ideal - differential polynomial ideal criterion - (optional) selection criterion; only works for Equations L - differential polynomial or a list or set of differential polynomials R - differential polynomial ring or ideal opts - (optional) sequence of options

Options

 • notation = jet, tjet, diff, or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of the first argument, ideal or L, is used.
 • fullset = true or false. This option only works with Equations and is incompatible with the option notation = diff. If fullset = true, Equations(ideal) also applies the function over the differential polynomials which state that the derivatives of the parameters are zero. The default value is false.
 • memout = nonnegative. Specifies a memory limit, in MB, for the computation. The default value is zero (no memory out).

Description

 • The commands Equations(ideal) and Inequations(ideal) return the equations and inequations of a differential ideal. The list of differential polynomials returned as the equations is assumed to be equal to zero and represents the chain of a regular differential chain. The list of differential polynomials returned as the inequations is assumed to be different from zero. In general, they are restrictions to the differential chain or case; specifically, they represent the initials and separants of the differential ideal.
 • The inequations of a regular differential chain also represent conditions that expansion points and initial values of power series solutions must satisfy. See the option conditions of PowerSeriesSolution.
 • To obtain the same equations returned by Equations but solved for the highest power of the leading derivative, pass the solved keyword anywhere in the sequence of arguments.
 • For both commands, if ideal is a list of regular differential chains, the commands are automatically mapped over the elements of the list.
 • The option criterion only works with Equations and restricts the equations returned to those satisfying the criterion specified (see below). When the first argument L is a list or set of differential polynomials, these two commands are automatically mapped over the elements of L; if criterion is omitted, L is returned.
 The possible values of criterion are listed here with the symbol * representing any one of the relational operators >, >=, <, <=, =, <>.
 – order * n where n is a nonnegative integer. In this case, Equations only returns differential polynomials whose orders satisfy the criterion. Differential polynomials, which do not depend on any derivative, are not selected.
 – rank * rk. The special ranks $0$ and $1$ are allowed. In this case, Equations only returns differential polynomials whose leading ranks satisfy the criterion.
 – leader * v, where v is a derivative. In this case, Equations only returns differential polynomials whose leading derivatives satisfy the criterion. This form applies also to differential polynomials which depend on independent variables only.
 – leader * derivative(v) where * represents either = or <> and v is a derivative. In this case, Equations only returns differential polynomials whose leading derivatives are equal to (or different from) any derivative of v. This form applies also to differential polynomials which depend on independent variables only.
 – leader * proper(v) where * represents either = or <> and v is a derivative. In this case, Equations only returns differential polynomials whose leading derivatives are equal to (or different from) any proper derivative of v. This form applies also to differential polynomials which depend on independent variables only.
 • This command is part of the DifferentialAlgebra package. It can be called using the form Equations(...) after executing the command with(DifferentialAlgebra). It can also be directly called using the form DifferentialAlgebra[Equations](...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):$
 > $R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{blocks}=\left[\left[u,v\right],p\right],\mathrm{parameters}=\left[p\right]\right)$
 ${R}{:=}{\mathrm{differential_ring}}$ (1)
 > $\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\left[{u}_{x}^{2}-4u,{u}_{x,y}{v}_{y}-u+p,{v}_{x,x}-{u}_{x}\right],R\right)$
 ${\mathrm{ideal}}{:=}\left[{\mathrm{regular_differential_chain}}{,}{\mathrm{regular_differential_chain}}\right]$ (2)

The two equations stating that the derivatives of $p$ are zero are omitted.

 > $\mathrm{Equations}\left(\mathrm{ideal}\right)$
 $\left[\left[{-}{p}{}{{v}}_{{x}{,}{x}}{+}{u}{}{{v}}_{{x}{,}{x}}{-}{2}{}{{u}}_{{y}}{}{{v}}_{{y}}{,}{-}{p}{}{{u}}_{{x}}{+}{u}{}{{u}}_{{x}}{-}{2}{}{{u}}_{{y}}{}{{v}}_{{y}}{,}{{u}}_{{y}}^{{2}}{-}{2}{}{u}{,}{-}{{p}}^{{2}}{+}{2}{}{p}{}{u}{-}{{u}}^{{2}}{+}{2}{}{{v}}_{{y}}^{{2}}\right]{,}\left[{{v}}_{{x}{,}{x}}{,}{u}{,}{p}\right]\right]$ (3)
 > $\mathrm{Equations}\left(\mathrm{ideal},\mathrm{solved}\right)$
 $\left[\left[{{v}}_{{x}{,}{x}}{=}\frac{{2}{}{{u}}_{{y}}{}{{v}}_{{y}}}{{u}{-}{p}}{,}{{u}}_{{x}}{=}\frac{{2}{}{{u}}_{{y}}{}{{v}}_{{y}}}{{u}{-}{p}}{,}{{u}}_{{y}}^{{2}}{=}{2}{}{u}{,}{{v}}_{{y}}^{{2}}{=}\frac{{1}}{{2}}{}{{u}}^{{2}}{-}{u}{}{p}{+}\frac{{1}}{{2}}{}{{p}}^{{2}}\right]{,}\left[{{v}}_{{x}{,}{x}}{=}{0}{,}{u}{=}{0}{,}{p}{=}{0}\right]\right]$ (4)
 > $\mathrm{Inequations}\left(\mathrm{ideal}\right)$
 $\left[\left[{{u}}_{{y}}{,}{{v}}_{{y}}{,}{u}{-}{p}\right]{,}\left[{}\right]\right]$ (5)

Compare with the initials and separants of $\mathrm{ideal}$

 > $\mathrm{Tools}:-\mathrm{Initial}\left(\mathrm{ideal}\right),\mathrm{Tools}:-\mathrm{Separant}\left(\mathrm{ideal}\right)$
 $\left[\left[{u}{-}{p}{,}{u}{-}{p}{,}{1}{,}{2}\right]{,}\left[{1}{,}{1}{,}{1}\right]\right]{,}\left[\left[{u}{-}{p}{,}{u}{-}{p}{,}{2}{}{{u}}_{{y}}{,}{4}{}{{v}}_{{y}}\right]{,}\left[{1}{,}{1}{,}{1}\right]\right]$ (6)

The two equations returned by Equations, stating that the derivatives of $p$ are zero, are present when using fullset = true or directly fullset

 > $\mathrm{Equations}\left({\mathrm{ideal}}_{1},\mathrm{fullset}\right)$
 $\left[{-}{p}{}{{v}}_{{x}{,}{x}}{+}{u}{}{{v}}_{{x}{,}{x}}{-}{2}{}{{u}}_{{y}}{}{{v}}_{{y}}{,}{-}{p}{}{{u}}_{{x}}{+}{u}{}{{u}}_{{x}}{-}{2}{}{{u}}_{{y}}{}{{v}}_{{y}}{,}{{u}}_{{y}}^{{2}}{-}{2}{}{u}{,}{-}{{p}}^{{2}}{+}{2}{}{p}{}{u}{-}{{u}}^{{2}}{+}{2}{}{{v}}_{{y}}^{{2}}{,}{{p}}_{{x}}{,}{{p}}_{{y}}\right]$ (7)
 > $\mathrm{Equations}\left({\mathrm{ideal}}_{1},\mathrm{fullset},\mathrm{solved}\right)$
 $\left[{{v}}_{{x}{,}{x}}{=}\frac{{2}{}{{u}}_{{y}}{}{{v}}_{{y}}}{{u}{-}{p}}{,}{{u}}_{{x}}{=}\frac{{2}{}{{u}}_{{y}}{}{{v}}_{{y}}}{{u}{-}{p}}{,}{{u}}_{{y}}^{{2}}{=}{2}{}{u}{,}{{v}}_{{y}}^{{2}}{=}\frac{{1}}{{2}}{}{{u}}^{{2}}{-}{u}{}{p}{+}\frac{{1}}{{2}}{}{{p}}^{{2}}{,}{{p}}_{{x}}{=}{0}{,}{{p}}_{{y}}{=}{0}\right]$ (8)

Selects the equations of order greater than $1$ only.

 > $\mathrm{Equations}\left(\mathrm{ideal},1<\mathrm{order}\right)$
 $\left[\left[{-}{p}{}{{v}}_{{x}{,}{x}}{+}{u}{}{{v}}_{{x}{,}{x}}{-}{2}{}{{u}}_{{y}}{}{{v}}_{{y}}\right]{,}\left[{{v}}_{{x}{,}{x}}\right]\right]$ (9)
 > $\mathrm{Equations}\left(\mathrm{ideal},1<\mathrm{order},\mathrm{solved}\right)$
 $\left[\left[{{v}}_{{x}{,}{x}}{=}\frac{{2}{}{{u}}_{{y}}{}{{v}}_{{y}}}{{u}{-}{p}}\right]{,}\left[{{v}}_{{x}{,}{x}}{=}{0}\right]\right]$ (10)

Selects the equations of rank less than or equal to the special rank $1$, i.e., all the numeric differential polynomials.

 > $\mathrm{Equations}\left(\left[0,17,{u}_{x}\right],\mathrm{rank}\le 1,R\right)$
 $\left[{17}{,}{0}\right]$ (11)

Note that for Equations to accomplish its task, the differential polynomial must include one of the dependent or independent variables of the differential ring. Otherwise, an error occurs.

Selects the equations whose leading derivatives are derivatives of $u$.

 > $\mathrm{Equations}\left({\mathrm{ideal}}_{1},\mathrm{leader}=\mathrm{derivative}\left(u\right)\right)$
 $\left[{-}{p}{}{{u}}_{{x}}{+}{u}{}{{u}}_{{x}}{-}{2}{}{{u}}_{{y}}{}{{v}}_{{y}}{,}{{u}}_{{y}}^{{2}}{-}{2}{}{u}\right]$ (12)
 > $\mathrm{Equations}\left({\mathrm{ideal}}_{1},\mathrm{leader}=\mathrm{derivative}\left(u\right),\mathrm{solved}\right)$
 $\left[{{u}}_{{x}}{=}\frac{{2}{}{{u}}_{{y}}{}{{v}}_{{y}}}{{u}{-}{p}}{,}{{u}}_{{y}}^{{2}}{=}{2}{}{u}\right]$ (13)
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 35, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9544380307197571, "perplexity": 1467.719537366353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276305.39/warc/CC-MAIN-20160524002116-00164-ip-10-185-217-139.ec2.internal.warc.gz"}