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.

$\mathrm{FunctionAdvisor}\left(\mathrm{DE}\,\mathrm{GaussAGM}\right)$

$\left[f\left(x\,y\right)=\mathrm{GaussAGM}\left(x\,y\right)\,\left[\frac{{\partial }^3}{\partial y^3}f\left(x\,y\right)=\left(\frac{6\left(\frac{\partial }{\partial y}f\left(x\,y\right)\right)}{f\left(x\,y\right)}+\frac{-x^4+9x^2{y}^2-8y^4}{y{\left(-y+x\right)}^2{\left(x+y\right)}^2}\right)\left(\frac{{\partial }^2}{\partial y^2}f\left(x\,y\right)\right)-\frac{6{\left(\frac{\partial }{\partial y}f\left(x\,y\right)\right)}^3}{{f\left(x\,y\right)}^2}+\frac{\left(2x^4-18x^2{y}^2+16y^4\right){\left(\frac{\partial }{\partial y}f\left(x\,y\right)\right)}^2}{y{\left(-y+x\right)}^2{\left(x+y\right)}^{2}f\left(x\,y\right)}+\frac{\left(x^4+6x^2{y}^2-13y^4\right)\left(\frac{\partial }{\partial y}f\left(x\,y\right)\right)}{y^2{\left(-y+x\right)}^2{\left(x+y\right)}^2}+\frac{3yf\left(x\,y\right)}{{\left(-y+x\right)}^2{\left(x+y\right)}^2}\,\frac{\partial }{\partial x}f\left(x\,y\right)=-\frac{y\left(\frac{\partial }{\partial y}f\left(x\,y\right)\right)}{x}+\frac{f\left(x\,y\right)}{x}\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{ODE}\,\mathrm{KelvinBei}\right)$

$\left[f\left(z\right)=\mathrm{KelvinBei}\left(a\,z\right)\,\left[\frac{{\ⅆ}^4}{\ⅆz^4}f\left(z\right)=-\frac{2\left(\frac{{\ⅆ}^3}{\ⅆz^3}f\left(z\right)\right)}{z}+\frac{\left(2a^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆz^2}f\left(z\right)\right)}{z^2}+\frac{\left(-2a^2-1\right)\left(\frac{\ⅆ}{\ⅆz}f\left(z\right)\right)}{z^3}+\frac{\left(-a^4-z^4+4a^2\right)f\left(z\right)}{z^4}\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{ODE}\,\mathrm{KelvinBei}\right)$

$\left[f\left(z\right)=\mathrm{KelvinBei}\left(a\,z\right)\,\left[\frac{{\ⅆ}^4}{\ⅆz^4}f\left(z\right)=-\frac{2\left(\frac{{\ⅆ}^3}{\ⅆz^3}f\left(z\right)\right)}{z}+\frac{\left(2a^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆz^2}f\left(z\right)\right)}{z^2}+\frac{\left(-2a^2-1\right)\left(\frac{\ⅆ}{\ⅆz}f\left(z\right)\right)}{z^3}+\frac{\left(-a^4-z^4+4a^2\right)f\left(z\right)}{z^4}\right]\right]$

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

$\mathrm{false}$

$\mathrm{FunctionAdvisor}\left(\mathrm{ODE}\,\mathrm{KelvinBei}\left(a\,z\right)\,\left[z\right]\right)$

$\left[f\left(z\right)=\mathrm{KelvinBei}\left(a\,z\right)\,\left[\frac{{\ⅆ}^4}{\ⅆz^4}f\left(z\right)=-\frac{2\left(\frac{{\ⅆ}^3}{\ⅆz^3}f\left(z\right)\right)}{z}+\frac{\left(2a^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆz^2}f\left(z\right)\right)}{z^2}+\frac{\left(-2a^2-1\right)\left(\frac{\ⅆ}{\ⅆz}f\left(z\right)\right)}{z^3}+\frac{\left(-a^4-z^4+4a^2\right)f\left(z\right)}{z^4}\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{DE}\,\mathrm{JacobiTheta1}\left(a\,z\right)\right)$

$\left[f\left(a\,z\right)=\mathrm{JacobiTheta1}\left(a\,z\right)\,\left[\frac{{\partial }^2}{\partial a^2}f\left(a\,z\right)=-4\left(\frac{\partial }{\partial z}f\left(a\,z\right)\right)z\right]\right]$

$\mathrm{depends}\left(\left[\right]\,a\right),\mathrm{depends}\left(\left[\right]\,z\right)$

$\mathrm{true},\mathrm{true}$

$\mathrm{JacobiTheta1}\left(a\,\frac{1}{\ln \left(z\right)}\right)$

$\mathrm{JacobiTheta1}\left(a\,\frac{1}{\ln \left(z\right)}\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{DE}\,\,\left[a\,z\right]\right)$

$\left[f\left(a\,z\right)=\mathrm{JacobiTheta1}\left(a\,\frac{1}{\ln \left(z\right)}\right)\,\left[\frac{{\partial }^3}{\partial a^2\partial z}f\left(a\,z\right)=\left(\frac{\frac{{\partial }^2}{\partial z^2}f\left(a\,z\right)}{\frac{\partial }{\partial z}f\left(a\,z\right)}+\frac{1}{z}\right)\left(\frac{{\partial }^2}{\partial a^2}f\left(a\,z\right)\right)+4\frac{\partial }{\partial z}f\left(a\,z\right)\,\frac{{\partial }^3}{\partial a^3}f\left(a\,z\right)=\frac{\left(\frac{{\partial }^2}{\partial a^2}f\left(a\,z\right)\right)\left(\frac{{\partial }^2}{\partial a\partial z}f\left(a\,z\right)\right)}{\frac{\partial }{\partial z}f\left(a\,z\right)}\right]\right]$

$\mathrm{JacobiTheta1}\left(a\,\frac{1}{\ln \left(z\right)}\right)$

$\mathrm{JacobiTheta1}\left(a\,\frac{1}{\ln \left(z\right)}\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{DE}\,\,g\left(a\,z\right)\right)$

$\left[g\left(a\,z\right)=\mathrm{JacobiTheta1}\left(a\,\frac{1}{\ln \left(z\right)}\right)\,\left[\frac{{\partial }^3}{\partial a^2\partial z}g\left(a\,z\right)=\left(\frac{\frac{{\partial }^2}{\partial z^2}g\left(a\,z\right)}{\frac{\partial }{\partial z}g\left(a\,z\right)}+\frac{1}{z}\right)\left(\frac{{\partial }^2}{\partial a^2}g\left(a\,z\right)\right)+4\frac{\partial }{\partial z}g\left(a\,z\right)\,\frac{{\partial }^3}{\partial a^3}g\left(a\,z\right)=\frac{\left(\frac{{\partial }^2}{\partial a^2}g\left(a\,z\right)\right)\left(\frac{{\partial }^2}{\partial a\partial z}g\left(a\,z\right)\right)}{\frac{\partial }{\partial z}g\left(a\,z\right)}\right]\right]$