return the differential equation form of a given mathematical function

Parameters

 DE - name where DE is one of the literal names 'DE', 'ODE', or 'PDE' math_function - Maple name of mathematical function var - optional, a list of independent variable(s) or the main dependent variable

Description

 • The FunctionAdvisor(DE, math_function) command returns an all polynomial differential equation system satisfied by the function, when it exists.
 A differential equation system is polynomial when it is polynomial in the independent variables, unknown functions, and its derivatives. For more information, see dpolyform.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{DE},\mathrm{GaussAGM}\right)$
 $\left[{f}{}\left({x}{,}{y}\right){=}{\mathrm{GaussAGM}}{}\left({x}{,}{y}\right){,}\left[\frac{{{\partial }}^{{3}}}{{\partial }{{y}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right){=}\left(\frac{{6}{}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right)}{{f}{}\left({x}{,}{y}\right)}{+}\frac{{-}{{x}}^{{4}}{+}{9}{}{{x}}^{{2}}{}{{y}}^{{2}}{-}{8}{}{{y}}^{{4}}}{{y}{}{\left({-}{y}{+}{x}\right)}^{{2}}{}{\left({x}{+}{y}\right)}^{{2}}}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){-}\frac{{6}{}{\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right)}^{{3}}}{{{f}{}\left({x}{,}{y}\right)}^{{2}}}{+}\frac{\left({2}{}{{x}}^{{4}}{-}{18}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{16}{}{{y}}^{{4}}\right){}{\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right)}^{{2}}}{{y}{}{\left({-}{y}{+}{x}\right)}^{{2}}{}{\left({x}{+}{y}\right)}^{{2}}{}{f}{}\left({x}{,}{y}\right)}{+}\frac{\left({{x}}^{{4}}{+}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{-}{13}{}{{y}}^{{4}}\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right)}{{{y}}^{{2}}{}{\left({-}{y}{+}{x}\right)}^{{2}}{}{\left({x}{+}{y}\right)}^{{2}}}{+}\frac{{3}{}{y}{}{f}{}\left({x}{,}{y}\right)}{{\left({-}{y}{+}{x}\right)}^{{2}}{}{\left({x}{+}{y}\right)}^{{2}}}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right){=}{-}\frac{{y}{}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right)}{{x}}{+}\frac{{f}{}\left({x}{,}{y}\right)}{{x}}\right]\right]$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{ODE},\mathrm{KelvinBei}\right)$
 $\left[{f}{}\left({z}\right){=}{\mathrm{KelvinBei}}{}\left({a}{,}{z}\right){,}\left[\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{z}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right){=}{-}\frac{{2}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{z}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{z}}{+}\frac{\left({2}{}{{a}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{{z}}^{{2}}}{+}\frac{\left({-}{2}{}{{a}}^{{2}}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{{z}}^{{3}}}{+}\frac{\left({-}{{a}}^{{4}}{-}{{z}}^{{4}}{+}{4}{}{{a}}^{{2}}\right){}{f}{}\left({z}\right)}{{{z}}^{{4}}}\right]\right]$ (2)

Note that 'ODE' and 'PDE' are synonyms for 'DE'.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{ODE},\mathrm{KelvinBei}\right)$
 $\left[{f}{}\left({z}\right){=}{\mathrm{KelvinBei}}{}\left({a}{,}{z}\right){,}\left[\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{z}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right){=}{-}\frac{{2}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{z}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{z}}{+}\frac{\left({2}{}{{a}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{{z}}^{{2}}}{+}\frac{\left({-}{2}{}{{a}}^{{2}}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{{z}}^{{3}}}{+}\frac{\left({-}{{a}}^{{4}}{-}{{z}}^{{4}}{+}{4}{}{{a}}^{{2}}\right){}{f}{}\left({z}\right)}{{{z}}^{{4}}}\right]\right]$ (3)

The variables used by the FunctionAdvisor command to create the calling sequence are local variables. Therefore, the previous examples does not depend on z.

 > $\mathrm{depends}\left(\left[\right],z\right)$
 ${\mathrm{false}}$ (4)

To make the FunctionAdvisor command return results using global variables, pass the actual function call instead of the function name. In this case, for some special functions it is also necessary to indicate which variables are the "differential equation independent variables". For examples, consider the following.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{ODE},\mathrm{KelvinBei}\left(a,z\right),\left[z\right]\right)$
 $\left[{f}{}\left({z}\right){=}{\mathrm{KelvinBei}}{}\left({a}{,}{z}\right){,}\left[\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{z}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right){=}{-}\frac{{2}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{z}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{z}}{+}\frac{\left({2}{}{{a}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{{z}}^{{2}}}{+}\frac{\left({-}{2}{}{{a}}^{{2}}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{{z}}^{{3}}}{+}\frac{\left({-}{{a}}^{{4}}{-}{{z}}^{{4}}{+}{4}{}{{a}}^{{2}}\right){}{f}{}\left({z}\right)}{{{z}}^{{4}}}\right]\right]$ (5)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{DE},\mathrm{JacobiTheta1}\left(a,z\right)\right)$
 $\left[{f}{}\left({a}{,}{z}\right){=}{\mathrm{JacobiTheta1}}{}\left({a}{,}{z}\right){,}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{a}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right){=}{-}{4}{}\left(\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right)\right){}{z}\right]\right]$ (6)
 > $\mathrm{depends}\left(\left[\right],a\right),\mathrm{depends}\left(\left[\right],z\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (7)

Moreover, in the case of the DE topic, the arguments in the function call need not be a name, in which case the differential equation system for the composed function call given is computed. Compare the differential equation for JacobiTheta1 above and this other one taking $\frac{1}{\mathrm{ln}\left(z\right)}$ as argument

 > $\mathrm{JacobiTheta1}\left(a,\frac{1}{\mathrm{ln}\left(z\right)}\right)$
 ${\mathrm{JacobiTheta1}}{}\left({a}{,}\frac{{1}}{{\mathrm{ln}}{}\left({z}\right)}\right)$ (8)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{DE},,\left[a,z\right]\right)$
 $\left[{f}{}\left({a}{,}{z}\right){=}{\mathrm{JacobiTheta1}}{}\left({a}{,}\frac{{1}}{{\mathrm{ln}}{}\left({z}\right)}\right){,}\left[\frac{{{\partial }}^{{3}}}{{\partial }{{a}}^{{2}}{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right){=}\left(\frac{\frac{{{\partial }}^{{2}}}{{\partial }{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right)}{\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right)}{+}\frac{{1}}{{z}}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{a}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right)\right){+}{4}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right){,}\frac{{{\partial }}^{{3}}}{{\partial }{{a}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right){=}\frac{\left(\frac{{{\partial }}^{{2}}}{{\partial }{{a}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right)\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{a}{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right)\right)}{\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{z}\right)}\right]\right]$ (9)

Apart from optionally passing a list of independent variables, one can pass the dependent variable directly (it includes the information on the independent variables):

 > $\mathrm{JacobiTheta1}\left(a,\frac{1}{\mathrm{ln}\left(z\right)}\right)$
 ${\mathrm{JacobiTheta1}}{}\left({a}{,}\frac{{1}}{{\mathrm{ln}}{}\left({z}\right)}\right)$ (10)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{DE},,g\left(a,z\right)\right)$
 $\left[{g}{}\left({a}{,}{z}\right){=}{\mathrm{JacobiTheta1}}{}\left({a}{,}\frac{{1}}{{\mathrm{ln}}{}\left({z}\right)}\right){,}\left[\frac{{{\partial }}^{{3}}}{{\partial }{{a}}^{{2}}{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right){=}\left(\frac{\frac{{{\partial }}^{{2}}}{{\partial }{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right)}{\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right)}{+}\frac{{1}}{{z}}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{a}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right)\right){+}{4}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right){,}\frac{{{\partial }}^{{3}}}{{\partial }{{a}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right){=}\frac{\left(\frac{{{\partial }}^{{2}}}{{\partial }{{a}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right)\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{a}{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right)\right)}{\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{g}{}\left({a}{,}{z}\right)}\right]\right]$ (11)
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