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

Consider the following differential ring, where the dependent variables are {p(x), q, u(x, y), v(x, y)}. As explained in DifferentialRing when entering the dependent variables, the dependency can optionally be indicated at the same time and it is necessary only for functions that do not depend on derivations, so you enter it as follows:
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$R\u2254\mathrm{DifferentialRing}\left(\mathrm{derivations}\=\left[x\,y\right]\,\mathrm{blocks}\=\left[\left[v\,u\right]\,q\left(\right)\,p\left(x\right)\right]\,\mathrm{arbitrary}\=p\right)$

${R}{\u2254}{\mathrm{differential\_ring}}$
 (2) 
The following example illustrates the Get command with the all the keywords related to getting differential ring information. Start with the ranking that contains the most relevant piece of information: the independent and dependent variables and how are they ranked.
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$\mathrm{Get}\left(\mathrm{ran}\,R\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}ran\text{'}\; against\; keyword\; \text{'}ranking\text{'}}}$
 
${\mathrm{derivations}}{\=}\left[{x}{\,}{y}\right]{\,}{\mathrm{blocks}}{\=}\left[\left[{v}{\,}{u}\right]{\,}{q}{\,}{p}\right]$
 (3) 
The following example shows all of the differential ring data.
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$\mathrm{Get}\left(\mathrm{data}\,R\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}data\text{'}\; against\; keyword\; \text{'}differentialringdata\text{'}}}$
 
${\mathrm{derivations}}{\=}\left[{x}{\,}{y}\right]{\,}{\mathrm{blocks}}{\=}\left[\left[{v}{\,}{u}\right]{\,}{q}{\,}{p}\right]{\,}{\mathrm{parameters}}{\=}\left[{p}{}\left({x}\right){\,}{q}\right]{\,}{\mathrm{arbitrary}}{\=}\left[{p}\right]$
 (4) 
The following example shows only the independent variables. There are two ways of getting the independent variables: the first method is useful for redefining differential rings (see the examples in DifferentialRing).
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$\mathrm{Get}\left(\mathrm{ions}\,R\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}ions\text{'}\; against\; keyword\; \text{'}derivations\text{'}}}$
 
${\mathrm{derivations}}{\=}\left[{x}{\,}{y}\right]$
 (5) 
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$\mathrm{Get}\left(\mathrm{ind}\,R\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}ind\text{'}\; against\; keyword\; \text{'}independentvariables\text{'}}}$
 
$\left\{{x}{\,}{y}\right\}$
 (6) 
The following example shows only the dependent variables. The first two are useful for redefining differential rings
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$\mathrm{Get}\left(\mathrm{blocks}\,R\right)$

${\mathrm{blocks}}{\=}\left[\left[{v}{\,}{u}\right]{\,}{q}{\,}{p}\right]$
 (7) 
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$\mathrm{Get}\left(\mathrm{blockswith}\,R\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}blockswith\text{'}\; against\; keyword\; \text{'}blockswithdependency\text{'}}}$
 
${\mathrm{blocks}}{\=}\left[\left[{v}{}\left({x}{\,}{y}\right){\,}{u}{}\left({x}{\,}{y}\right)\right]{\,}{q}{}\left({}\right){\,}{p}{}\left({x}\right)\right]$
 (8) 
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$\mathrm{Get}\left(\mathrm{dep}\,R\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}dep\text{'}\; against\; keyword\; \text{'}dependentvariables\text{'}}}$
 
$\left\{{p}{}\left({x}\right){\,}{q}{}\left({}\right){\,}{u}{}\left({x}{\,}{y}\right){\,}{v}{}\left({x}{\,}{y}\right)\right\}$
 (9) 
The following examples show only the arbitrary equations and the parameters. This output can also be used as is for redefining differential rings.
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$\mathrm{Get}\left(\mathrm{arbitrary}\,R\right)$

${\mathrm{arbitrary}}{\=}\left[{p}\right]$
 (10) 
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$\mathrm{Get}\left(\mathrm{arbitrarywith}\,R\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}arbitrarywith\text{'}\; against\; keyword\; \text{'}arbitrarywithdependency\text{'}}}$
 
${\mathrm{arbitrary}}{\=}\left[{p}{}\left({x}\right)\right]$
 (11) 
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$\mathrm{Get}\left(p\,R\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}p\text{'}\; against\; keyword\; \text{'}parameters\text{'}}}$
 
${\mathrm{parameters}}{\=}\left[{p}{}\left({x}\right){\,}{q}\right]$
 (12) 
Consider the following differential equation system, and the ideal returned for it by RosenfeldGroebner using R as differential ring.
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$\mathrm{ee}\u2254u\left(x\,y\right)q\+v\left(x\,y\right)p\left(x\right)\:$

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$\mathrm{sys}\u2254\left(\frac{{\partial}^{2}}{\partial y\partial x}\mathrm{ee}\right)\mathrm{ee}\,\frac{{\partial}^{2}}{\partial {y}^{2}}\left({\mathrm{ee}}^{2}\right)$

${\mathrm{sys}}{\u2254}\left(\left(\frac{{{\partial}}^{{2}}}{{\partial}{y}{}{\partial}{x}}{}{u}{}\left({x}{\,}{y}\right)\right){}{q}{\+}\left(\frac{{{\partial}}^{{2}}}{{\partial}{y}{}{\partial}{x}}{}{v}{}\left({x}{\,}{y}\right)\right){}{p}{}\left({x}\right){\+}\left(\frac{{\partial}}{{\partial}{y}}{}{v}{}\left({x}{\,}{y}\right)\right){}\left(\frac{{\ⅆ}}{{\ⅆ}{x}}{}{p}{}\left({x}\right)\right)\right){}\left({u}{}\left({x}{\,}{y}\right){}{q}{\+}{v}{}\left({x}{\,}{y}\right){}{p}{}\left({x}\right)\right){\,}{2}{}{\left(\left(\frac{{\partial}}{{\partial}{y}}{}{u}{}\left({x}{\,}{y}\right)\right){}{q}{\+}\left(\frac{{\partial}}{{\partial}{y}}{}{v}{}\left({x}{\,}{y}\right)\right){}{p}{}\left({x}\right)\right)}^{{2}}{\+}{2}{}\left({u}{}\left({x}{\,}{y}\right){}{q}{\+}{v}{}\left({x}{\,}{y}\right){}{p}{}\left({x}\right)\right){}\left(\left(\frac{{{\partial}}^{{2}}}{{\partial}{{y}}^{{2}}}{}{u}{}\left({x}{\,}{y}\right)\right){}{q}{\+}\left(\frac{{{\partial}}^{{2}}}{{\partial}{{y}}^{{2}}}{}{v}{}\left({x}{\,}{y}\right)\right){}{p}{}\left({x}\right)\right)$
 (13) 
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$\mathrm{ideal}\u2254\mathrm{RosenfeldGroebner}\left(\left[\mathrm{sys}\right]\,R\right)$

${\mathrm{ideal}}{\u2254}\left[{\mathrm{regular\_differential\_chain}}{\,}{\mathrm{regular\_differential\_chain}}\right]$
 (14) 
Thus, this ideal has two differential chains or, in other words, there are two cases: the general and one singular. You can view information about this ideal using Get.
The following example shows the ring embedded in the ideal.
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$\mathrm{recovered\_ring}\u2254\mathrm{Get}\left(\mathrm{differentialring}\,\mathrm{ideal}\right)$

${\mathrm{recovered\_ring}}{\u2254}{\mathrm{differential\_ring}}$
 (15) 
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$\mathrm{Get}\left(\mathrm{data}\,\mathrm{recovered\_ring}\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}data\text{'}\; against\; keyword\; \text{'}differentialringdata\text{'}}}$
 
${\mathrm{derivations}}{\=}\left[{x}{\,}{y}\right]{\,}{\mathrm{blocks}}{\=}\left[\left[{v}{\,}{u}\right]{\,}{q}{\,}{p}\right]{\,}{\mathrm{parameters}}{\=}\left[{p}{}\left({x}\right){\,}{q}\right]{\,}{\mathrm{arbitrary}}{\=}\left[{p}\right]$
 (16) 
Note the attributes of each ideal: both are prime.
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$\mathrm{Get}\left(\mathrm{at}\,\mathrm{ideal}\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}at\text{'}\; against\; keyword\; \text{'}attributes\text{'}}}$
 
$\left[\left[{\mathrm{differential}}{\,}{\mathrm{prime}}{\,}{\mathrm{autoreduced}}{\,}{\mathrm{primitive}}{\,}{\mathrm{squarefree}}{\,}{\mathrm{coherent}}{\,}{\mathrm{normalized}}\right]{\,}\left[{\mathrm{differential}}{\,}{\mathrm{prime}}{\,}{\mathrm{autoreduced}}{\,}{\mathrm{primitive}}{\,}{\mathrm{squarefree}}{\,}{\mathrm{coherent}}{\,}{\mathrm{normalized}}\right]\right]$
 (17) 
This example shows the differential order of each case.
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$\mathrm{Get}\left(\mathrm{order}\,\mathrm{ideal}\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}order\text{'}\; against\; keyword\; \text{'}differentialorder\text{'}}}$
 
$\left[{1}{\,}{2}\right]$
 (18) 
this example shows the derivatives in each differential chain (in each case); note that derivatives of order zero are also displayed.
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$\mathrm{Get}\left(\mathrm{derivatives}\,\mathrm{ideal}\right)$

$\left[\left\{{q}{\,}\frac{{\partial}}{{\partial}{y}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{\partial}}{{\partial}{y}}{}{v}{}\left({x}{\,}{y}\right){\,}{p}{}\left({x}\right)\right\}{\,}\left\{{q}{\,}\frac{{\ⅆ}}{{\ⅆ}{x}}{}{p}{}\left({x}\right){\,}\frac{{\partial}}{{\partial}{x}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{\partial}}{{\partial}{y}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{{y}}^{{2}}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{\partial}}{{\partial}{x}}{}{v}{}\left({x}{\,}{y}\right){\,}\frac{{\partial}}{{\partial}{y}}{}{v}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{{y}}^{{2}}}{}{v}{}\left({x}{\,}{y}\right){\,}{p}{}\left({x}\right){\,}{u}{}\left({x}{\,}{y}\right){\,}{v}{}\left({x}{\,}{y}\right)\right\}\right]$
 (19) 
The derivatives of order zero are the dependent variables themselves and can also be obtained by passing a ring instead of an ideal.
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$\mathrm{Get}\left(\mathrm{derivatives}\,R\right)$

$\left\{{q}{\,}{p}{}\left({x}\right){\,}{u}{}\left({x}{\,}{y}\right){\,}{v}{}\left({x}{\,}{y}\right)\right\}$
 (20) 
The following example shows the derivatives up to order 2 that will require a value as initial conditions when computing formal power series solution up to order 2.
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$\mathrm{Get}\left(\mathrm{initial}\,2\,\mathrm{ideal}\right)$

${\mathrm{*\; Partial\; match\; of\; \text{'}initial\text{'}\; against\; keyword\; \text{'}initialconditions\text{'}}}$
 
$\left[\left\{{q}{\,}\frac{{\ⅆ}}{{\ⅆ}{x}}{}{p}{}\left({x}\right){\,}\frac{{{\ⅆ}}^{{2}}}{{\ⅆ}{{x}}^{{2}}}{}{p}{}\left({x}\right){\,}\frac{{\partial}}{{\partial}{x}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{\partial}}{{\partial}{y}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{{x}}^{{2}}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{{y}}^{{2}}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{\partial}}{{\partial}{x}}{}{v}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{{x}}^{{2}}}{}{v}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{y}{}{\partial}{x}}{}{u}{}\left({x}{\,}{y}\right){\,}{p}{}\left({x}\right){\,}{u}{}\left({x}{\,}{y}\right){\,}{v}{}\left({x}{\,}{y}\right)\right\}{\,}\left\{{q}{\,}\frac{{\ⅆ}}{{\ⅆ}{x}}{}{p}{}\left({x}\right){\,}\frac{{{\ⅆ}}^{{2}}}{{\ⅆ}{{x}}^{{2}}}{}{p}{}\left({x}\right){\,}\frac{{\partial}}{{\partial}{x}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{\partial}}{{\partial}{y}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{{x}}^{{2}}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{{y}}^{{2}}}{}{u}{}\left({x}{\,}{y}\right){\,}\frac{{\partial}}{{\partial}{y}}{}{v}{}\left({x}{\,}{y}\right){\,}\frac{{{\partial}}^{{2}}}{{\partial}{y}{}{\partial}{x}}{}{u}{}\left({x}{\,}{y}\right){\,}{p}{}\left({x}\right){\,}{u}{}\left({x}{\,}{y}\right){\,}{v}{}\left({x}{\,}{y}\right)\right\}\right]$
 (21) 
The following example shows dependent variables that involve fewer independent variables than those declared in derivations (in this example, $x\,y$); in the framework of DifferentialAlgebra, these dependent variables are also called parameters.
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$\mathrm{Get}\left(\mathrm{parameters}\,\mathrm{ideal}\right)$

${\mathrm{parameters}}{\=}\left[{p}{}\left({x}\right){\,}{q}\right]$
 (22) 
The following example shows the notation used in these differential chains.
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$\mathrm{Get}\left(\mathrm{notation}\,\mathrm{ideal}\right)$
