Updates to Differential Equation (DE) Solvers in Maple 16 - Maple Programming Help

Updates to Differential Equation (DE) Solvers in Maple 16

Summary

 • The theme for exact, symbolic Ordinary Differential Equation solving in Maple 16 is the development of new solving methods for 1st, 2nd, and higher order nonlinear ODEs. The new methods extend the ability to solving 1st order Abel and other families of equations, and a number of 2nd and higher order families of equations not admitting point symmetries, or admitting them but in cases where they are of no practical use. The Maple 16 ODE solver keeps pushing the frontier in exact solutions for ODEs. Maple remains the state of the art in computing exact solutions for ODEs with computer algebra software.
 – For 1st order ODEs, dsolve can additionally solve two new 1-parameter Abel ODE classes and two other entire new ODE classes depending on one arbitrary function that happen to be linearizable through integral transformations.
 – For each of the 2nd, 3rd, and 4th order ODEs problems, in Maple 16 dsolve can additionally solve two new nonlinear ODE families depending on arbitrary functions.
 Recalling, each new solvable class of equations includes infinitely many different equations that can be transformed into each other through the transformations of the dependent and independent variables that define the class.
 • For both Ordinary and Partial Differential Equations, a theme in Maple 16 is a thorough revision of the existing symmetry algorithms and their extension to automatically handle problems involving anticommutative variables. This extension includes the dsolve and pdsolve commands, all the commands of the PDEtools[Library] and many of the symmetry commands in PDEtools, making all this DE functionality be naturally available for the first time for working on problems in Physics that involve anticommutative variables.

Ordinary Differential Equations (ODEs)

 Using new algorithms, the dsolve command can additionally solve new nonlinear ODE families of order 1st, 2nd, 3rd and 4th, all of them parametrized by arbitrary functions of the independent variable and out of reach in previous Maple releases.

New solvable 1st order nonlinear ODE families

 • For 1st order ODEs, the simplest problem beyond the reach of complete solving algorithms is known as Abel equations. These are equations of the form

$\mathrm{y\text{'}}=\frac{{y}^{3}{f}_{3}+{y}^{2}{f}_{2}+y{f}_{1}+{f}_{0}}{y{g}_{1}+{g}_{0}}$

 where $y\equiv y\left(x\right)$ is the unknown and the ${f}_{i}\equiv {f}_{i}\left(x\right)$ and ${g}_{j}\equiv {g}_{j}\left(x\right)$ are arbitrary functions of $x$. The biggest subclass of Abel equations known to be solvable was discovered by our research team and is the AIR 4-parameter class. New in Maple 16, two additional 1-parameter classes of Abel equations, beyond the AIR class, are now also solvable.
 Examples
 > PDEtools[declare](y(x), prime=x);
 ${y}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{y}$
 ${\mathrm{derivatives with respect to}}{}{x}{}{\mathrm{of functions of one variable will now be displayed with \text{'}}}$ (1)
 This equation, of type Abel 2nd kind, depending on one parameter $\mathrm{alpha}$, is now solved in terms of hypergeometric functions
 > ode[1] := diff(y(x),x) = -y(x)*(-3+x)*((-9+2*alpha)*y(x)-9)/(x*(54-36*x+6*x^2-3*alpha+2*alpha*x)*y(x)+9*alpha+81-54*x+9*x^2);
 ${{\mathrm{ode}}}_{{1}}{≔}{\mathrm{y\text{'}}}{=}{-}\frac{{y}{}\left({-}{3}{+}{x}\right){}\left(\left({-}{9}{+}{2}{}{\mathrm{α}}\right){}{y}{-}{9}\right)}{{x}{}\left({2}{}{\mathrm{α}}{}{x}{+}{6}{}{{x}}^{{2}}{-}{3}{}{\mathrm{α}}{-}{36}{}{x}{+}{54}\right){}{y}{+}{9}{}{\mathrm{α}}{+}{81}{-}{54}{}{x}{+}{9}{}{{x}}^{{2}}}$ (2)
 > dsolve(ode[1]);
 ${\mathrm{_C1}}{+}\frac{{27}}{{2}}{}\frac{{\left({x}{}{y}{+}{3}\right)}^{{2}}{}\left({{y}}^{{4}}{}\left({\mathrm{α}}{+}\frac{{9}}{{4}}\right){}\left({\mathrm{α}}{+}{9}\right){}{\mathrm{α}}{}{\mathrm{LerchPhi}}{}\left({-}\frac{{2}}{{9}}{}{y}{}{\mathrm{α}}{+}{y}{+}{1}{,}{1}{,}\frac{{8}{}{\mathrm{α}}{-}{9}}{{-}{9}{+}{2}{}{\mathrm{α}}}\right){+}\frac{{243}}{{8}}{+}\left({-}\frac{{3}}{{2}}{}{{\mathrm{α}}}^{{2}}{-}\frac{{135}}{{8}}{}{\mathrm{α}}{-}\frac{{243}}{{8}}\right){}{{y}}^{{3}}{+}\left(\frac{{27}}{{8}}{}{\mathrm{α}}{+}\frac{{243}}{{8}}\right){}{{y}}^{{2}}{-}\frac{{243}}{{8}}{}{y}\right){}{\left(\frac{{2}}{{9}}{}{y}{}{\mathrm{α}}{-}{y}{-}{1}\right)}^{\frac{{8}{}{\mathrm{α}}{-}{9}}{{-}{9}{+}{2}{}{\mathrm{α}}}}{-}\frac{{81}}{{2}}{}{\left(\frac{{2}}{{9}}{}{y}{}{\mathrm{α}}{-}{y}{-}{1}\right)}^{\frac{{6}{}{\mathrm{α}}}{{-}{9}{+}{2}{}{\mathrm{α}}}}{}\left({-}\frac{{9}}{{2}}{+}\left({\mathrm{α}}{-}\frac{{9}}{{2}}\right){}{y}\right){}\left(\frac{{3}}{{2}}{+}\left({x}{-}\frac{{3}}{{2}}\right){}{y}\right)}{{\left({\mathrm{α}}{-}\frac{{9}}{{2}}\right)}^{{4}}{}{{y}}^{{4}}{}{\left({x}{}{y}{+}{3}\right)}^{{2}}}{=}{0}$ (3)
 The related class of Abel equations that is now entirely solvable consists of the set of equations that can be obtained from equation (2) by changing variables

$x→F\left(x\right),y→\frac{P[1]\left(x\right)y+Q[1]\left(x\right)}{P[2]\left(x\right)y+Q[2]\left(x\right)}$

 where $G\left(x\right)$ and the four $P[i]\left(x\right),Q[j]\left(x\right)$ are arbitrary rational functions of $x$; this is the most general transformation that preserves the form of Abel equations and thus generates Abel ODE classes.
 • The following ODE family depending on an arbitrary function $G\left(x\right)$ is representative of the next difficult problem beyond Abel equations, that is, a problem involving 4th powers of $y\left(x\right)$ in the right-hand side, now under focus
 > ode[2] := diff(y(x),x) = 1/2*(G(x)^2*y(x)^4-diff(G(x),x)*y(x)^2+x)/G(x)/y(x);
 ${{\mathrm{ode}}}_{{2}}{≔}{\mathrm{y\text{'}}}{=}\frac{{1}}{{2}}{}\frac{{{G}{}\left({x}\right)}^{{2}}{}{{y}}^{{4}}{-}{\mathrm{G\text{'}}}{}{{y}}^{{2}}{+}{x}}{{G}{}\left({x}\right){}{y}}$ (4)
 • We solve it here in implicit form to avoid square roots obscuring the solution
 > sol[2] := dsolve(ode[2], y(x), implicit);
 ${{\mathrm{sol}}}_{{2}}{≔}{\mathrm{_C1}}{+}\frac{{\mathrm{AiryBi}}{}\left({-}{x}\right){}{G}{}\left({x}\right){}{{y}}^{{2}}{-}{\mathrm{AiryBi}}{}\left({1}{,}{-}{x}\right)}{{\mathrm{AiryAi}}{}\left({-}{x}\right){}{G}{}\left({x}\right){}{{y}}^{{2}}{-}{\mathrm{AiryAi}}{}\left({1}{,}{-}{x}\right)}{=}{0}$ (5)
 > odetest(sol[2], ode[2], y(x));
 ${0}$ (6)
 • A generalization of the problem above, solvable in Maple 16, involving an arbitrary function $G\left(y\left(x\right)\right)$
 > ode[3] := diff(y(x),x) = 2*G(y(x))*x/(2-D(G)(y(x))*x^2+G(y(x))^2*x^4-2*y(x)*G(y(x))*x^2);
 ${{\mathrm{ode}}}_{{3}}{≔}{\mathrm{y\text{'}}}{=}\frac{{2}{}{G}{}\left({y}\right){}{x}}{{2}{-}{\mathrm{D}}{}\left({G}\right){}\left({y}\right){}{{x}}^{{2}}{+}{{G}{}\left({y}\right)}^{{2}}{}{{x}}^{{4}}{-}{2}{}{y}{}{G}{}\left({y}\right){}{{x}}^{{2}}}$ (7)
 > sol[3] := dsolve(ode[3]);
 ${{\mathrm{sol}}}_{{3}}{≔}{\mathrm{_C1}}{+}\frac{{-}{G}{}\left({y}\right){}{{x}}^{{2}}{+}{2}{}{y}}{{-}{2}{}{{ⅇ}}^{{{y}}^{{2}}}{+}\left({-}{G}{}\left({y}\right){}{{x}}^{{2}}{+}{2}{}{y}\right){}{\mathrm{erfi}}{}\left({y}\right){}\sqrt{{\mathrm{π}}}}{=}{0}$ (8)
 > odetest(sol[3], ode[3]);
 ${0}$ (9)

New solvable nonlinear ODE families of 2nd, 3rd, and 4th order

 Using new algorithms developed by our research team, the dsolve command in Maple 16 can additionally solve two new nonlinear ODE families for each of the 2nd, 3rd and 4th order problems, with the ODE families involving arbitrary functions of the independent ($x$) or dependent ($y\left(x\right)$) variables.
 Examples
 • A 4th order ODE family
 > ode[4] := diff(y(x), x$4) = 5*y(x)*diff(y(x), x$3) + (10*diff(y(x), x) - 10*y(x)^2-1/x)*diff(y(x), x$2) - 15*y(x)*diff(y(x), x)^2+(10*y(x)^3 + 3/x*y(x))*diff(y(x),x) - y(x)^5 - 1/x*y(x)^3;  ${{\mathrm{ode}}}_{{4}}{≔}{\mathrm{y\text{'}\text{'}\text{'}\text{'}}}{=}{5}{}{y}{}{\mathrm{y\text{'}\text{'}\text{'}}}{+}\left({10}{}{\mathrm{y\text{'}}}{-}{10}{}{{y}}^{{2}}{-}\frac{{1}}{{x}}\right){}{\mathrm{y\text{'}\text{'}}}{-}{15}{}{y}{}{{\mathrm{y\text{'}}}}^{{2}}{+}\left({10}{}{{y}}^{{3}}{+}\frac{{3}{}{y}}{{x}}\right){}{\mathrm{y\text{'}}}{-}{{y}}^{{5}}{-}\frac{{{y}}^{{3}}}{{x}}$ (10)  This ODE has no point symmetries; the determining PDE for the symmetry infinitesimals only admits both of them equal to zero:  > PDEtools:-DeterminingPDE(ode[4]);  $\left\{{{\mathrm{_η}}}_{{y}}{}\left({x}{,}{y}\right){=}{0}{,}{{\mathrm{_ξ}}}_{{x}}{}\left({x}{,}{y}\right){=}{0}\right\}$ (11)  • Using new algorithms, this problem is nevertheless solvable in explicit form in terms of Bessel functions  > dsolve(ode[4]);  ${y}{=}\frac{{-}{\mathrm{BesselJ}}{}\left({3}{,}{2}{}\sqrt{{x}}\right){}{{x}}^{{3}{/}{2}}{}{\mathrm{_C4}}{-}{\mathrm{BesselY}}{}\left({3}{,}{2}{}\sqrt{{x}}\right){}{{x}}^{{3}{/}{2}}{-}{2}{}{\mathrm{_C3}}{}{x}{-}{\mathrm{_C2}}}{{\mathrm{_C1}}{+}{\mathrm{_C2}}{}{x}{+}{\mathrm{_C3}}{}{{x}}^{{2}}{+}{\mathrm{_C4}}{}{{x}}^{{2}}{}{\mathrm{BesselJ}}{}\left({4}{,}{2}{}\sqrt{{x}}\right){+}{{x}}^{{2}}{}{\mathrm{BesselY}}{}\left({4}{,}{2}{}\sqrt{{x}}\right)}$ (12)  • Another problem not admitting point symmetries, of 3rd order  > ode[5] := diff(y(x), x$3) = (4*y(x)-1/x)*diff(y(x), x$2) + 3*diff(y(x), x)^2 + (-6*y(x)^2+3/x*y(x)-x)*diff(y(x), x) + y(x)^4 - 1/x*y(x)^3 + y(x)^2*x - y(x);  ${{\mathrm{ode}}}_{{5}}{≔}{\mathrm{y\text{'}\text{'}\text{'}}}{=}\left({4}{}{y}{-}\frac{{1}}{{x}}\right){}{\mathrm{y\text{'}\text{'}}}{+}{3}{}{{\mathrm{y\text{'}}}}^{{2}}{+}\left({-}{6}{}{{y}}^{{2}}{+}\frac{{3}{}{y}}{{x}}{-}{x}\right){}{\mathrm{y\text{'}}}{+}{{y}}^{{4}}{-}\frac{{{y}}^{{3}}}{{x}}{+}{{y}}^{{2}}{}{x}{-}{y}$ (13)  > dsolve(ode[5]);  ${y}{=}\frac{{1}}{{240}}{}\frac{{-}{240}{}{\mathrm{_C2}}{}{x}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{3}}{,}\frac{{1}}{{3}}\right]{,}\left[\frac{{2}}{{3}}{,}\frac{{2}}{{3}}{,}\frac{{4}}{{3}}\right]{,}{-}\frac{{1}}{{9}}{}{{x}}^{{3}}\right){+}{15}{}{\mathrm{_C2}}{}{{x}}^{{4}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{4}}{{3}}{,}\frac{{4}}{{3}}\right]{,}\left[\frac{{5}}{{3}}{,}\frac{{5}}{{3}}{,}\frac{{7}}{{3}}\right]{,}{-}\frac{{1}}{{9}}{}{{x}}^{{3}}\right){-}{480}{}{\mathrm{_C3}}{}{{x}}^{{2}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{2}}{{3}}{,}\frac{{2}}{{3}}\right]{,}\left[{1}{,}\frac{{4}}{{3}}{,}\frac{{5}}{{3}}\right]{,}{-}\frac{{1}}{{9}}{}{{x}}^{{3}}\right){+}{16}{}{\mathrm{_C3}}{}{{x}}^{{5}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{5}}{{3}}{,}\frac{{5}}{{3}}\right]{,}\left[{2}{,}\frac{{7}}{{3}}{,}\frac{{8}}{{3}}\right]{,}{-}\frac{{1}}{{9}}{}{{x}}^{{3}}\right){-}{720}{}{\mathrm{MeijerG}}{}\left(\left[\left[{1}\right]{,}\left[{}\right]\right]{,}\left[\left[\frac{{2}}{{3}}{,}\frac{{2}}{{3}}{,}\frac{{1}}{{3}}\right]{,}\left[{}\right]\right]{,}\frac{{1}}{{9}}{}{{x}}^{{3}}\right)}{{x}{}\left({\mathrm{_C1}}{+}{\mathrm{_C2}}{}{x}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{3}}{,}\frac{{1}}{{3}}\right]{,}\left[\frac{{2}}{{3}}{,}\frac{{2}}{{3}}{,}\frac{{4}}{{3}}\right]{,}{-}\frac{{1}}{{9}}{}{{x}}^{{3}}\right){+}{\mathrm{_C3}}{}{{x}}^{{2}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{2}}{{3}}{,}\frac{{2}}{{3}}\right]{,}\left[{1}{,}\frac{{4}}{{3}}{,}\frac{{5}}{{3}}\right]{,}{-}\frac{{1}}{{9}}{}{{x}}^{{3}}\right){+}{\mathrm{MeijerG}}{}\left(\left[\left[{1}{,}{1}\right]{,}\left[{}\right]\right]{,}\left[\left[\frac{{2}}{{3}}{,}\frac{{2}}{{3}}{,}\frac{{1}}{{3}}\right]{,}\left[{0}\right]\right]{,}\frac{{1}}{{9}}{}{{x}}^{{3}}\right)\right)}$ (14)  • A 2nd order nonlinear ODE problem for which point symmetries exist but are of no use for integration purposes (they involve an unsolved 4th order linear ODE).  > ode[6] := diff(y(x), x, x) = x*(diff(y(x), x))^3*y(x)+x^2*(x-1)*(diff(y(x), x))^3+(-3*x+1)*(diff(y(x), x))^2;  ${{\mathrm{ode}}}_{{6}}{≔}{\mathrm{y\text{'}\text{'}}}{=}{x}{}{{\mathrm{y\text{'}}}}^{{3}}{}{y}{+}{{x}}^{{2}}{}\left({x}{-}{1}\right){}{{\mathrm{y\text{'}}}}^{{3}}{+}\left({-}{3}{}{x}{+}{1}\right){}{{\mathrm{y\text{'}}}}^{{2}}$ (15)  > sol[6] := dsolve(ode[6]);  ${{\mathrm{sol}}}_{{6}}{≔}\frac{\left({\mathrm{AiryAi}}{}\left({-}{y}{+}\frac{{1}}{{4}}\right){}{\mathrm{_C2}}{+}{\mathrm{AiryBi}}{}\left({-}{y}{+}\frac{{1}}{{4}}\right)\right){}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}{y}}}{{{∫}}_{{}}^{{y}}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}{\mathrm{_a}}}{}\left({\mathrm{AiryAi}}{}\left({-}{\mathrm{_a}}{+}\frac{{1}}{{4}}\right){}{\mathrm{_C2}}{+}{\mathrm{AiryBi}}{}\left({-}{\mathrm{_a}}{+}\frac{{1}}{{4}}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}{+}{x}{=}{0}$ (16)  > odetest(sol[6], ode[6]);  ${0}$ (17)  • A 3rd order ODE problem illustrating improvements in existing algorithms: in previous releases this equation was only solved in terms of uncomputed integrals of unresolved RootOf expressions; now the solution is computed explicitly as a rational function  > ode[7] := diff(y(x), x$3) + 4*y(x)*diff(y(x), x\$2) + 3*diff(y(x), x)^2 + 6*y(x)^2*diff(y(x), x) + y(x)^4 = 0;
 ${{\mathrm{ode}}}_{{7}}{≔}{\mathrm{y\text{'}\text{'}\text{'}}}{+}{4}{}{y}{}{\mathrm{y\text{'}\text{'}}}{+}{3}{}{{\mathrm{y\text{'}}}}^{{2}}{+}{6}{}{\mathrm{y\text{'}}}{}{{y}}^{{2}}{+}{{y}}^{{4}}{=}{0}$ (18)
 > sol[7] := dsolve(ode[7]);
 ${{\mathrm{sol}}}_{{7}}{≔}{y}{=}\frac{{3}{}{\mathrm{_C1}}{}{{x}}^{{2}}{+}{6}{}{\mathrm{_C2}}{}{x}{+}{6}{}{\mathrm{_C3}}}{{\mathrm{_C1}}{}{{x}}^{{3}}{+}{3}{}{\mathrm{_C2}}{}{{x}}^{{2}}{+}{6}{}{\mathrm{_C3}}{}{x}{+}{6}}$ (19)
 > odetest(sol[7], ode[7]);
 ${0}$ (20)
 • dsolve can now solve ODEs that involve anticommutative variables set using the Physics package using the approach explained in PerformOnAnticommutativeSystem.
 > with(Physics, Setup);
 $\left[{\mathrm{Setup}}\right]$ (21)
 • Set first $\mathrm{theta}$ and $Q$ as suffixes for variables of type/anticommutative (see Physics[Setup])
 > Setup(anticommutativepre = {theta, Q});
 ${\mathrm{* Partial match of \text{'}anticommutativepre\text{'} against keyword \text{'}anticommutativeprefix\text{'}}}$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{Q}{,}{\mathrm{_λ}}{,}{\mathrm{θ}}\right\}\right]$ (22)
 • Consider this ordinary differential equation for the anticommutative function $Q$ of a commutative variable $x$
 > diff(Q(x), x, x) - Q(x)*diff(Q(x), x) = 0;
 ${\mathrm{Q\text{'}\text{'}}}{-}{Q}{}\left({x}\right){}{\mathrm{Q\text{'}}}{=}{0}$ (23)
 • Its solution using dsolve involves an anticommutative constant $\mathrm{_lambda1}$, analogous to the commutative constants $\mathrm{_C1}$
 > dsolve((23));
 ${Q}{}\left({x}\right){=}{\mathrm{tan}}{}\left(\frac{{1}}{{2}}{}\sqrt{{\mathrm{_C1}}{}{\mathrm{_λ1}}}{}\left({\mathrm{_C2}}{+}{x}\right){}\sqrt{{2}}\right){}\sqrt{{\mathrm{_C1}}{}{\mathrm{_λ1}}}{}\sqrt{{2}}$ (24)

Partial Differential Equations (PDEs)

 Many of the commands in PDEtools can now naturally handle anticommutative variables, these are: D_Dx, DeterminingPDE, dsubs, Eta_k, FromJet, FunctionFieldSolutions, InfinitesimalGenerator, Infinitesimals, InvariantSolutions, PolynomialSolutions, ReducedForm, ToJet as well as all the routines within the PDEtools[Library]. This makes it possible to tackle super PDE problems, that is, PDE systems involving anticommutative functions and variables. In addition, this permits the use of the PDE mathematical tools with the Physics package. (Related to these developments, see also the new Physics commands Coefficients, ToFieldComponents, ToSuperfields and PerformOnAnticommutativeSystem.)
 During the development of this generalization of PDEtools commands to handle anticommutative variables, the symmetry ones were also improved in a number of places, all this setting a new benchmark for the state-of-the-art in symmetry analysis and computation of exact solutions for partial differential equations.

Examples

 • Both dsolve and pdsolve can now solve PDEs that involve anticommutative variables set using the Physics package using the approach explained in PerformOnAnticommutativeSystem.
 > $\mathrm{with}\left(\mathrm{PDEtools}\right),\mathrm{with}\left(\mathrm{Physics}\right):$
 Set first $\mathrm{theta}$ and $Q$ as suffixes for variables of type/anticommutative (see Physics[Setup])
 > $\mathrm{Setup}\left(\mathrm{anticommutativepre}=\left\{\mathrm{θ},Q\right\}\right)$
 ${\mathrm{* Partial match of \text{'}anticommutativepre\text{'} against keyword \text{'}anticommutativeprefix\text{'}}}$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{Q}{,}{\mathrm{_λ}}{,}{\mathrm{θ}}\right\}\right]$ (25)
 Consider this partial differential equation for the anticommutative function $Q$ of commutative and anticommutative variables $x,\mathrm{theta}$
 > $\frac{{\partial }^{2}}{\partial \mathrm{θ}\partial x}Q\left(x,y,\mathrm{θ}\right)=0$
 ${{Q}}_{{x}{,}{\mathrm{\theta }}}{=}{0}$ (26)
 Its solution using pdsolve
 > $\mathrm{pdsolve}\left(\right)$
 ${Q}{}\left({x}{,}{y}{,}{\mathrm{θ}}\right){=}{\mathrm{_F2}}{}\left({x}{,}{y}\right){}{\mathrm{_λ2}}{+}{\mathrm{_F4}}{}\left({y}\right){}{\mathrm{θ}}$ (27)
 Note the introduction of an anticommutative constant $\mathrm{_lambda2}$, analogous to the commutative constants $\mathrm{_Cn}$ where n is an integer. The arbitrary functions $\mathrm{_Fn}$ introduced are all commutative as usual and the Grassmannian parity (on right-hand-side if compared with the one on the left-hand-side) is preserved
 > $\mathrm{Physics}:-\mathrm{GrassmannParity}\left(\right)$
 ${1}{=}{1}$ (28)
 A PDE system example with one unknown anticommutative function $Q$ of four variables, two commutative and two anticommutative; to avoid redundant typing in the input that follows and redundant display of information on the screen, use PDEtools:-declare, and PDEtools:-diff_table, that also handles anticommutative variables by automatically using Physics:-diff when Physics is loaded
 > $\mathrm{PDEtools}:-\mathrm{declare}\left(Q\left(x,y,{\mathrm{θ}}_{1},{\mathrm{θ}}_{2}\right)\right)$
 ${Q}{}\left({x}{,}{y}{,}{{\mathrm{θ}}}_{{1}}{,}{{\mathrm{θ}}}_{{2}}\right){}{\mathrm{will now be displayed as}}{}{Q}$ (29)
 > $q≔\mathrm{PDEtools}:-\mathrm{diff_table}\left(Q\left(x,y,{\mathrm{θ}}_{1},{\mathrm{θ}}_{2}\right)\right):$
 Now we can enter derivatives directly as the function's name indexed by the differentiation variables and see the display the same way; two PDEs
 > ${\mathrm{pde}}_{1}≔{q}_{x,y,{\mathrm{θ}}_{1}}+{q}_{x,y,{\mathrm{θ}}_{2}}-{q}_{y,{\mathrm{θ}}_{1},{\mathrm{θ}}_{2}}=0$
 ${{\mathrm{pde}}}_{{1}}{≔}{{Q}}_{{y}{,}{x}{,}{{\mathrm{\theta }}}_{{1}}}{+}{{Q}}_{{y}{,}{x}{,}{{\mathrm{\theta }}}_{{2}}}{-}{{Q}}_{{{\mathrm{\theta }}}_{{1}}{,}{y}{,}{{\mathrm{\theta }}}_{{2}}}{=}{0}$ (30)
 > ${\mathrm{pde}}_{2}≔{q}_{{\mathrm{θ}}_{1}}=0$
 ${{\mathrm{pde}}}_{{2}}{≔}{{Q}}_{{{\mathrm{\theta }}}_{{1}}}{=}{0}$ (31)
 The solution to this system:
 > $\mathrm{pdsolve}\left(\left[{\mathrm{pde}}_{1},{\mathrm{pde}}_{2}\right]\right)$
 ${Q}{=}{\mathrm{_F4}}{}\left({x}{,}{y}\right){}{\mathrm{_λ3}}{+}\left({\mathrm{_F9}}{}\left({x}\right){+}{\mathrm{_F8}}{}\left({y}\right)\right){}{{\mathrm{θ}}}_{{2}}$ (32)
 • The dsubs command also works with anticommutative variables, though natively without using the approach explained in PerformOnAnticommutativeSystem. By inspection, it is clear that the derivatives in pde[2] can be substituted in pde[1] reducing the problem to a simpler one:
 > $\mathrm{dsubs}\left({\mathrm{pde}}_{2},{\mathrm{pde}}_{1}\right)$
 ${{Q}}_{{y}{,}{x}{,}{{\mathrm{\theta }}}_{{2}}}{=}{0}$ (33)
 > $\mathrm{pdsolve}\left(\right)$
 ${Q}{=}{\mathrm{_F4}}{}\left({x}{,}{y}\right){}{\mathrm{_λ3}}{+}{\mathrm{_F5}}{}\left({x}{,}{y}\right){}{{\mathrm{θ}}}_{{1}}{+}\left({\mathrm{_F9}}{}\left({x}\right){+}{\mathrm{_F8}}{}\left({y}\right)\right){}{{\mathrm{θ}}}_{{2}}{+}\left({\mathrm{_F11}}{}\left({x}\right){+}{\mathrm{_F10}}{}\left({y}\right)\right){}{\mathrm{_λ4}}{}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}$ (34)
 Substituting this result for $Q$ back into pde[2], then multiplying by ${\mathrm{theta}}_{1}$ and subtracting from the above also leads to the PDE system solution.
 Using differential elimination techniques and the approach explained in PerformOnAnticommutativeSystem, ReducedForm in this example arrives at the same result as dsubs
 > $\mathrm{PDEtools}:-\mathrm{ReducedForm}\left({\mathrm{pde}}_{1},{\mathrm{pde}}_{2}\right)$
 $\left[{{Q}}_{{y}{,}{x}{,}{{\mathrm{\theta }}}_{{2}}}\right]{&where}\left[{}\right]$