The dpolyform function receives an equation or expression, or a set or list of them, containing some unknown functions of one or many variables, possibly containing non-polynomial objects (fractional powers, mathematical functions, etc.) and returns an equivalent differential polynomial (DP) system of equations.

The DP returned is equivalent to the given system in that any solution (value of the unknown functions) of the input is also a solution of the output DP system. This DP returned is differential polynomial in that it is entirely polynomial in the unknowns and its derivatives, with coefficients entirely rational in the independent variables. In this sense, dpolyform has the concrete ability to compute differential polynomial representations for non-polynomial objects.

The mathematical expressions dpolyform can represent in DP form include almost all the functions of the mathematical language, arbitrary compositions of them with the operations $+,,\^,\@$, including in that abstract powers, inverse functions constructed with @@, functions represented as ODE solutions using DESol, algebraic expressions represented using RootOf, integrals represented with Int, int or Intat, etc. The only exceptions are mathematical functions related to Zeta (e.g. Psi and GAMMA), not admitting a polynomial differential equation representation.

This routine has enormous potential in that it permits the use of the existing methods for polynomial systems, for example, differential elimination (see, for example, DifferentialAlgebra, DEtools[Rif], or casesplit), to solve, or in general work with, non-polynomial systems in a systematic way.

The dpolyform command can be used to also decouple non-polynomial systems of differential or algebraic equations (unlike the output of casesplit, the output of dpolyform is DP), or in the computation of identities for special functions. The dpolyform subroutines are used by the Maple dsolve, pdsolve and casesplit commands when solving ODE and PDE systems, in order to triangularize non-polynomial systems using techniques for polynomial ones.

By default, dpolyform is expected to return rather fast, outputting a DP system equivalent to the input system, representing the non-polynomial objects by means of auxiliary functions $\mathrm{\_Fn}\left(...\right)$ (n is an integer) which in turn satisfy DP equations. Thus, the output of dpolyform consists of three lists.

-  The first list contains DP equations of the form "DP_expression = 0".

- The second list consists of inequations of the form "DP_expression <> 0".

- The third list consists of 'back substitution equations', showing which non-polynomial object is represented by each auxiliary function $\mathrm{\_Fn}$ introduced.

By giving the optional argument 'no_Fn', a second process is run to remove all the $\mathrm{\_Fn}$ functions introduced in the first step, ending with a DP system (which can be a single DP equation) that involves only the unknown functions found in the given input. This system that is free of auxiliary functions is obtained by running a differential elimination process ranking all the auxiliary $\mathrm{\_Fn}$ higher than the unknown functions present in the input. For details on rankings, see DEtools[checkrank].

The dpolyform routine also accepts the same optional arguments that are accepted by casesplit.

$\mathrm{with}\left(\mathrm{PDEtools}\right)\&colon;$

$\mathrm{declare}\left(g\left(x\&comma;y\right)\&comma;\mathrm{\_F1}\left(x\&comma;y\right)\&comma;\mathrm{\_F2}\left(x\&comma;y\right)\&comma;\mathrm{\_F3}\left(x\&comma;y\right)\right)$

$g\left(x\&comma;y\right)\mathrm{will\; now\; be\; displayed\; as}g$

$\mathrm{\_F1}\left(x\&comma;y\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F1}$

$\mathrm{\_F2}\left(x\&comma;y\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F2}$

$\mathrm{\_F3}\left(x\&comma;y\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F3}$

$\mathrm{e1}≔g\left(x\&comma;y\right)=\tan \left(2x-y^{\frac{1}{2}}\right)$

$\mathrm{e1}≔g=\tan \left(2x-\sqrt{y}\right)$

$\mathrm{DPS}≔\mathrm{dpolyform}\left(\mathrm{e1}\right)$

$\mathrm{DPS}≔\left[g-\mathrm{\_F1}=0\&comma;-2{\mathrm{\_F1}}_y\mathrm{\_F2}-1-{\mathrm{\_F1}}^2=0\&comma;-2{\mathrm{\_F1}}^2+{\mathrm{\_F1}}_x-2=0\&comma;{\mathrm{\_F2}}^2-y=0\&comma;{\mathrm{\_F2}}_x=0\right],\left[\mathrm{\_F2}\ne 0\&comma;{\mathrm{\_F1}}_x\ne 0\&comma;{\mathrm{\_F1}}_y\mathrm{\_F2}\ne 0\right],\left[\mathrm{\_F2}=\sqrt{y}\&comma;\mathrm{\_F1}=\tan \left(2x-\sqrt{y}\right)\right]$

$\mathrm{show}$

$\left[g-\mathrm{\_F1}=0\&comma;-2\left(\frac{\partial \mathrm{\_F1}}{\partial y}\right)\mathrm{\_F2}-1-{\mathrm{\_F1}}^2=0\&comma;-2{\mathrm{\_F1}}^2+\frac{\partial \mathrm{\_F1}}{\partial x}-2=0\&comma;{\mathrm{\_F2}}^2-y=0\&comma;\frac{\partial \mathrm{\_F2}}{\partial x}=0\right],\left[\mathrm{\_F2}\ne 0\&comma;\frac{\partial \mathrm{\_F1}}{\partial x}\ne 0\&comma;\left(\frac{\partial \mathrm{\_F1}}{\partial y}\right)\mathrm{\_F2}\ne 0\right],\left[\mathrm{\_F2}=\sqrt{y}\&comma;\mathrm{\_F1}=\tan \left(2x-\sqrt{y}\right)\right]$

$\mathrm{dpolyform}\left(\mathrm{e1}\&comma;\mathrm{no\_Fn}\right)$

$\left[g_x=2g^2+2\&comma;{g_y}^2=\frac{g^4}{4y}+\frac{g^2}{2y}+\frac{1}{4y}\right]\mathbf{where}\left[g_y\ne 0\&comma;-g^2-1\ne 0\right]$

$\mathrm{lprint}\left(\right)$

`casesplit/ans`([diff(g(x,y),x) = 2*g(x,y)^2+2, diff(g(x,y),y)^2 = 1/4/y*g(x,y)^4+1/2/y*g(x,y)^2+1/4/y],[diff(g(x,y),y) <> 0, -g(x,y)^2-1 <> 0])

$\mathrm{pde\_sys}≔\left[\mathrm{op}\left(\mathrm{map}\left(\mathrm{op}\&comma;\right)\right)\right]$

$\mathrm{pde\_sys}≔\left[g_x=2g^2+2\&comma;{g_y}^2=\frac{g^4}{4y}+\frac{g^2}{2y}+\frac{1}{4y}\&comma;g_y\ne 0\&comma;-g^2-1\ne 0\right]$

$\left\{\mathrm{e1}\right\}$

$\left\{g=\tan \left(2x-\sqrt{y}\right)\right\}$

$\mathrm{pdetest}\left(\&comma;\mathrm{pde\_sys}\right)$

$\left[0\&comma;0\right]$

$\mathrm{pdsolve}\left(\mathrm{pde\_sys}\right)$

$\left\{g=\tan \left(2x+\sqrt{y}+2\mathrm{c\_\_1}\right)\right\}$

$\mathrm{pde\_sys}\left[1..2\right]$

$\left[g_x=2g^2+2\&comma;{g_y}^2=\frac{g^4}{4y}+\frac{g^2}{2y}+\frac{1}{4y}\right]$

$\mathrm{pdsolve}\left(\right)$

$\left\{g=\mathrm{-I}\right\},\left\{g=\mathrm{I}\right\},\left\{g=\tan \left(2x+\sqrt{y}+2\mathrm{c\_\_1}\right)\right\}$

$\mathrm{declare}\left(f\left(x\right)\&comma;y\left(x\right)\&comma;\mathrm{prime}=x\right)$

$f\left(x\right)\mathrm{will\; now\; be\; displayed\; as}f$

$y\left(x\right)\mathrm{will\; now\; be\; displayed\; as}y$

$\mathrm{derivatives\; with\; respect\; to}x\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

$y\left(x\right)=\mathrm{BesselJ}\left(n\&comma;x\right)$

$y=\mathrm{BesselJ}\left(n\&comma;x\right)$

$\mathrm{dpolyform}\left(\&comma;\mathrm{no\_Fn}\right)$

$\left[\mathrm{y\text{'}\text{'}}=-\frac{\mathrm{y\text{'}}}{x}+\frac{\left(n^2-x^2\right)y}{x^2}\right]\mathbf{where}\left[y\ne 0\right]$

$y\left(x\right)=\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;f\left(x\right)\right)$

$y=\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;f\right)$

$\mathrm{dpolyform}\left(\&comma;\left[y\&comma;f\right]\&comma;\mathrm{no\_Fn}\right)$

$\left[\mathrm{y\text{'}\text{'}}=\frac{\mathrm{y\text{'}}\mathrm{f\; \text{'}\text{'}}}{\mathrm{f\; \text{'}}}-\frac{yab{\mathrm{f\; \text{'}}}^2}{f^2-f}+\frac{\left(\left(-a-b-1\right)f+c\right)\mathrm{y\text{'}}\mathrm{f\; \text{'}}}{f^2-f}\right]\mathbf{where}\left[y\ne 0\&comma;\mathrm{f\; \text{'}}\ne 0\right]$

$y\left(x\right)=\mathrm{MeijerG}\left(\left[\left[a\right]\&comma;\left[b\right]\right]\&comma;\left[\left[c\right]\&comma;\left[d\right]\right]\&comma;f\left(x\right)\right)$

$y=\mathrm{MeijerG}\left(\left[\left[a\right]\&comma;\left[b\right]\right]\&comma;\left[\left[c\right]\&comma;\left[d\right]\right]\&comma;f\right)$

$\mathrm{dpolyform}\left(\&comma;y\left(x\right)\&comma;\mathrm{no\_Fn}\right)$

$\left[\mathrm{y\text{'}\text{'}}=\frac{\mathrm{y\text{'}}\mathrm{f\; \text{'}\text{'}}}{\mathrm{f\; \text{'}}}+\frac{\left(-\left(a-1\right)\left(b-1\right)yf+ycd\right){\mathrm{f\; \text{'}}}^2}{f^3-f^2}+\frac{\left(\left(a+b-3\right)f^2+\left(-c-d+1\right)f\right)\mathrm{y\text{'}}\mathrm{f\; \text{'}}}{f^3-f^2}\right]\mathbf{where}\left[y\ne 0\&comma;\mathrm{f\; \text{'}}\ne 0\right]$

$\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;f\left(x\right)\right)=\mathrm{convert}\left(\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;f\left(x\right)\right)\&comma;\mathrm{MeijerG}\right)$

$\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;f\right)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{MeijerG}\left(\left[\left[-a+1\&comma;-b+1\right]\&comma;\left[\right]\right]\&comma;\left[\left[0\right]\&comma;\left[1-c\right]\right]\&comma;-f\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}$

$P\left(\mathrm{\nu }\&comma;k\right)=\mathrm{EllipticPi}\left(\mathrm{\nu }\&comma;k\right)$

$P\left(\mathrm{\nu }\&comma;k\right)=\mathrm{EllipticPi}\left(\mathrm{\nu }\&comma;k\right)$

$\mathrm{dpolyform}\left(\&comma;\mathrm{no\_Fn}\right)$

$\left[P_{k,k}=\frac{\left(-3k^4+\left(\mathrm{\nu }+1\right)k^2+\mathrm{\nu }\right)P_k}{k\left(k-1\right)\left(k+1\right)\left(k^2-\mathrm{\nu }\right)}+\frac{2\mathrm{\nu }\left(\mathrm{\nu }-1\right)P_{\mathrm{\nu }}}{\left(k-1\right)\left(k+1\right)\left(k^2-\mathrm{\nu }\right)}-\frac{P\left(\mathrm{\nu }\&comma;k\right)}{\left(k-1\right)\left(k+1\right)}\&comma;P_{k,\mathrm{\nu }}=\frac{P_k}{k^2-\mathrm{\nu }}-\frac{k{P}_{\mathrm{\nu }}}{k^2-\mathrm{\nu }}\&comma;P_{\mathrm{\nu },\mathrm{\nu }}=\frac{\left(-k^3+k\right)P_k}{2\mathrm{\nu }\left(\mathrm{\nu }-1\right)\left(k^2-\mathrm{\nu }\right)}+\frac{\left(-4\mathrm{\nu }{k}^2+k^2+5{\mathrm{\nu }}^2-2\mathrm{\nu }\right)P_{\mathrm{\nu }}}{2\mathrm{\nu }\left(\mathrm{\nu }-1\right)\left(k^2-\mathrm{\nu }\right)}-\frac{P\left(\mathrm{\nu }\&comma;k\right)}{2\mathrm{\nu }\left(\mathrm{\nu }-1\right)}\right]\mathbf{where}\left[P\left(\mathrm{\nu }\&comma;k\right)\ne 0\right]$

$y\left(x\right)={\mathrm{dawson}}^{\left(-1\right)}\left(x\right)$

$y={\mathrm{dawson}}^{\left(\mathrm{-1}\right)}\left(x\right)$

$\mathrm{dpolyform}\left(\&comma;\mathrm{no\_Fn}\right)$

$\left[\mathrm{y\text{'}}=-\frac{1}{2yx-1}\right]\mathbf{where}\left[y\ne 0\right]$

$\mathrm{DEtools}\left[\mathrm{odeadvisor}\right]\left(\mathrm{op}\left(\left[1\&comma;1\right]\&comma;\right)\right)$

$\left[\left[\mathrm{\_1st\_order}\&comma;\mathrm{\_with\_symmetry\_[F(x)*G(y),0]}\right]\&comma;\left[\mathrm{\_Abel}\&comma;\mathrm{2nd\; type}\&comma;\mathrm{class\; C}\right]\right]$

$\mathrm{a3}≔y\left(x\right)=\mathrm{hypergeom}\left(\left[1\right]\&comma;\left[2\right]\&comma;-2\mathrm{I}x\right)$

$\mathrm{a3}≔y=\mathrm{hypergeom}\left(\left[1\right]\&comma;\left[2\right]\&comma;-2\mathrm{I}x\right)$

$\mathrm{e3}≔\mathrm{dpolyform}\left(\mathrm{a3}\&comma;\mathrm{no\_Fn}\right)$

$\mathrm{e3}≔\left[\mathrm{y\text{'}\text{'}}=\frac{\left(-2\mathrm{I}x-2\right)\mathrm{y\text{'}}}{x}-\frac{2\mathrm{I}y}{x}\right]\mathbf{where}\left[y\ne 0\right]$

$\mathrm{op}\left(\left[1\&comma;1\right]\&comma;\mathrm{e3}\right)$

$\mathrm{y\text{'}\text{'}}=\frac{\left(-2\mathrm{I}x-2\right)\mathrm{y\text{'}}}{x}-\frac{2\mathrm{I}y}{x}$

$\mathrm{a3\_bis}≔\mathrm{dsolve}\left(\&comma;\left[\mathrm{Kovacic}\right]\right)$

$\mathrm{a3\_bis}≔y=\frac{\mathrm{c\_\_1}}{x}+\frac{\mathrm{c\_\_2}{\&ExponentialE;}^{-2\mathrm{I}x}}{x}$

$\mathrm{e4}≔\mathrm{a3}-\mathrm{a3\_bis}$

$\mathrm{e4}≔0=\mathrm{hypergeom}\left(\left[1\right]\&comma;\left[2\right]\&comma;-2\mathrm{I}x\right)-\frac{\mathrm{c\_\_1}}{x}-\frac{\mathrm{c\_\_2}{\&ExponentialE;}^{-2\mathrm{I}x}}{x}$

$\mathrm{series}\left(\mathrm{rhs}\left(\mathrm{e4}\right)\&comma;x\&comma;2\right)$

$\left(-\mathrm{c\_\_1}-\mathrm{c\_\_2}\right)x^{\mathrm{-1}}+1+2\mathrm{I}\mathrm{c\_\_2}+O\left(x\right)$

$\mathrm{sys}≔\left\{\mathrm{coeffs}\left(\mathrm{convert}\left(\&comma;\mathrm{polynom}\right)\&comma;x\right)\right\}$

$\mathrm{sys}≔\left\{1+2\mathrm{I}\mathrm{c\_\_2}\&comma;-\mathrm{c\_\_1}-\mathrm{c\_\_2}\right\}$

$\mathrm{ans\_C}≔\mathrm{solve}\left(\mathrm{sys}\&comma;\left\{\mathrm{\_C1}\&comma;\mathrm{\_C2}\right\}\right)$

$\mathrm{ans\_C}≔\left\{\mathrm{c\_\_1}=-\frac{\mathrm{I}}{2}\&comma;\mathrm{c\_\_2}=\frac{\mathrm{I}}{2}\right\}$

$\mathrm{eval}\left(\mathrm{e4}\&comma;\mathrm{ans\_C}\right)$

$0=\mathrm{hypergeom}\left(\left[1\right]\&comma;\left[2\right]\&comma;-2\mathrm{I}x\right)+\frac{\mathrm{I}}{2x}-\frac{\mathrm{I}{\&ExponentialE;}^{-2\mathrm{I}x}}{2x}$

$\mathrm{isolate}\left(\&comma;\mathrm{hypergeom}\left(\left[1\right]\&comma;\left[2\right]\&comma;-2\mathrm{I}x\right)\right)$

$\mathrm{hypergeom}\left(\left[1\right]\&comma;\left[2\right]\&comma;-2\mathrm{I}x\right)=-\frac{\mathrm{I}}{2x}+\frac{\mathrm{I}{\&ExponentialE;}^{-2\mathrm{I}x}}{2x}$

$\mathrm{convert}\left(\&comma;\mathrm{StandardFunctions}\right)$

$-\frac{\mathrm{I}}{2x}+\frac{\mathrm{I}{\&ExponentialE;}^{-2\mathrm{I}x}}{2x}=-\frac{\mathrm{I}}{2x}+\frac{\mathrm{I}{\&ExponentialE;}^{-2\mathrm{I}x}}{2x}$

$\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\right)$

$0$

$\mathrm{declare}\left(\left(y\&comma;z\right)\left(t\right)\&comma;\mathrm{prime}=t\right)$

$y\left(t\right)\mathrm{will\; now\; be\; displayed\; as}y$

$z\left(t\right)\mathrm{will\; now\; be\; displayed\; as}z$

$\mathrm{derivatives\; with\; respect\; to}t\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

$\mathrm{sys}≔\left[t-\tan \left(y\left(t\right)+z\left(t\right)-\ln \left(y\left(t\right)\right)\right)=0\&comma;y\left(t\right)-\exp \left(-y\left(t\right)+z\left(t\right)+\arctan \left(t\right)\right)=0\right]$

$\mathrm{sys}≔\left[t+\tan \left(-y-z+\ln \left(y\right)\right)=0\&comma;y-{\&ExponentialE;}^{-y+z+\arctan \left(t\right)}=0\right]$

$\mathrm{dpolyform}\left(\mathrm{sys}\&comma;\mathrm{no\_Fn}\right)$

$\left[\mathrm{y\text{'}}=\frac{1}{t^2+1}\&comma;\mathrm{z\text{'}}=\frac{1}{y\left(t^2+1\right)}\right]\mathbf{where}\left[y+1\ne 0\&comma;y\ne 0\right]$

$\mathrm{DP\_sys}≔\left[\mathrm{op}\left(\mathrm{map}\left(\mathrm{op}\&comma;\right)\right)\right]$

$\mathrm{DP\_sys}≔\left[\mathrm{y\text{'}}=\frac{1}{t^2+1}\&comma;\mathrm{z\text{'}}=\frac{1}{y\left(t^2+1\right)}\&comma;y+1\ne 0\&comma;y\ne 0\right]$

$\mathrm{sol\_DP\_sys}≔\mathrm{dsolve}\left(\mathrm{DP\_sys}\&comma;\mathrm{explicit}\right)$

$\mathrm{sol\_DP\_sys}≔\left\{y=\arctan \left(t\right)+\mathrm{c\_\_2}\&comma;z=\ln \left(\arctan \left(t\right)+\mathrm{c\_\_2}\right)+\mathrm{c\_\_1}\right\}$

$\mathrm{sys\_C}≔\mathrm{eval}\left(\mathrm{sys}\&comma;\mathrm{sol\_DP\_sys}\right)$

$\mathrm{sys\_C}≔\left[t-\tan \left(\arctan \left(t\right)+\mathrm{c\_\_2}+\mathrm{c\_\_1}\right)=0\&comma;\arctan \left(t\right)+\mathrm{c\_\_2}-{\&ExponentialE;}^{-\mathrm{c\_\_2}+\ln \left(\arctan \left(t\right)+\mathrm{c\_\_2}\right)+\mathrm{c\_\_1}}=0\right]$

$\mathrm{z1}≔\mathrm{map}\left(\mathrm{lhs}\&comma;\mathrm{sys\_C}\right)\&colon;$

$\mathrm{z2}≔\mathrm{map}\left(\mathrm{series}\&comma;\mathrm{z1}\&comma;t\&comma;1\right)\&colon;$

$\mathrm{z3}≔\mathrm{simplify}\left(\mathrm{map}\left(\mathrm{convert}\&comma;\mathrm{z2}\&comma;\mathrm{polynom}\right)\right)$

$\mathrm{z3}≔\left[-\tan \left(\mathrm{c\_\_2}+\mathrm{c\_\_1}\right)\&comma;\mathrm{c\_\_2}\left(1-{\&ExponentialE;}^{-\mathrm{c\_\_2}+\mathrm{c\_\_1}}\right)\right]$

$\mathrm{\_EnvAllSolutions}≔\mathrm{true}$

$\mathrm{\_EnvAllSolutions}≔\mathrm{true}$

$\mathrm{sol\_C}≔\mathrm{solve}\left(\left\{\mathrm{op}\left(\mathrm{z3}\right)\right\}\&comma;\left\{\mathrm{\_C1}\&comma;\mathrm{\_C2}\right\}\right)$

$\mathrm{sol\_C}≔\left\{\mathrm{c\_\_1}=\mathrm{\pi }\mathrm{\_Z1~}\&comma;\mathrm{c\_\_2}=0\right\},\left\{\mathrm{c\_\_1}=\frac{\mathrm{\pi }\mathrm{\_Z2~}}{2}+\mathrm{I}\mathrm{\pi }\mathrm{\_Z3~}\&comma;\mathrm{c\_\_2}=\frac{\mathrm{\pi }\mathrm{\_Z2~}}{2}-\mathrm{I}\mathrm{\pi }\mathrm{\_Z3~}\right\}$

$\mathrm{select}\left(\mathrm{has}\&comma;\left[\mathrm{sol\_C}\right]\&comma;\mathrm{I}\right)\left[1\right]$

$\left\{\mathrm{c\_\_1}=\frac{\mathrm{\pi }\mathrm{\_Z2~}}{2}+\mathrm{I}\mathrm{\pi }\mathrm{\_Z3~}\&comma;\mathrm{c\_\_2}=\frac{\mathrm{\pi }\mathrm{\_Z2~}}{2}-\mathrm{I}\mathrm{\pi }\mathrm{\_Z3~}\right\}$

$\mathrm{sol\_sys}≔\mathrm{eval}\left(\mathrm{sol\_DP\_sys}\&comma;\right)$

$\mathrm{sol\_sys}≔\left\{y=\arctan \left(t\right)+\frac{\mathrm{\pi }\mathrm{\_Z2~}}{2}-\mathrm{I}\mathrm{\pi }\mathrm{\_Z3~}\&comma;z=\ln \left(\arctan \left(t\right)+\frac{\mathrm{\pi }\mathrm{\_Z2~}}{2}-\mathrm{I}\mathrm{\pi }\mathrm{\_Z3~}\right)+\frac{\mathrm{\pi }\mathrm{\_Z2~}}{2}+\mathrm{I}\mathrm{\pi }\mathrm{\_Z3~}\right\}$

$\mathrm{sys}$

$\left[t+\tan \left(-y-z+\ln \left(y\right)\right)=0\&comma;y-{\&ExponentialE;}^{-y+z+\arctan \left(t\right)}=0\right]$

$\mathrm{simplify}\left(\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{sys}\&comma;\mathrm{sol\_sys}\right)\right)\right)$

$\left[0=0\&comma;0=0\right]$

Cheb-Terrab, E.S. "A Computational Approach for the Exact Solving of Systems of Partial Differential Equations." Submitted to Computer Physics Communications, 2001.