For the first three calling sequences, given a PDE, or a system of PDEs, possibly including ODEs, algebraic constraints, and inequations, the main goal of the pdsolve function is to find an analytical solution. There are no restrictions as to the type, differential order, or number of dependent or independent variables of the PDEs or PDE systems that pdsolve can try to solve.

For computing solutions to systems of partial differential equation see pdsolve/system.

For computing exact solutions to partial differential equations subject to boundary initial values see pdsolve/boundaryconditions and mode examples in in examples/pdsolve_boundaryconditions.

For computing series solutions to partial differential equation systems subject to initial values see pdsolve/series.

For the remaining (fourth) calling sequence above, pdsolve finds a numerical solution for the input PDE or PDE system. A discussion of this usage of pdsolve, the types of PDE that can be numerically solved, and any numeric-specific arguments can be found in pdsolve/numeric.

The remainder of this page discusses the exact solution of a single PDE. For help on the exact solution of PDE systems, see pdsolve/system.

Solving a single PDE

The pdsolve command currently recognizes a certain number of PDE families that can be solved by using standard methods. When the given PDE belongs to an unrecognized family, pdsolve uses a heuristic algorithm that attempts separation of variables based on the specific structure of the PDE. pdsolve also works with anticommutative variables set using the Physics package using the approach explained in PerformOnAnticommutativeSystem.

The strategy pdsolve uses is to look for the most general solution to the given PDE or, in the worst case, to look for a complete separation of variables.  Thus, when successful, the command returns one of the following:

- A general solution,

- A quasi-general solution (a solution containing arbitrary functions, but not in sufficient number or not having enough variables to constitute a general solution), or

- A set of uncoupled ODEs with all the variables separated, or a complete solution obtained after integrating this set (when the option INTEGRATE is indicated).

When an incomplete separation of variables is reached, the command calls itself again (with a smaller problem), possibly using different methods of solution. If the smaller problem cannot be solved, the incomplete separation of variables is returned (with a warning message).

The results of pdsolve are returned, by default, in one of three forms.

a) When the general solution to the PDE is obtained, pdsolve returns an explicit result for the indeterminate function in terms of arbitrary functions. For example, for $\mathrm{PDE}≔x\left(\frac{\partial }{\partial y}f\left(x,y\right)\right)-y\left(\frac{\partial }{\partial x}f\left(x,y\right)\right)=0$, the input $\mathrm{pdsolve}\left(\mathrm{PDE}\right)$ returns $f\left(x,y\right)=f_1\left(x^2+y^2\right)$, where $f_1$ is an arbitrary function.

b) When a solution, but not the most general one, is obtained pdsolve expresses the result using the internal PDESolStruc function, displayed as in 'solution where details', with solution being the functional form found for the indeterminate function, and details contains a list with any ODEs found while separating the variables, as well as any arbitrary functions or changes of variables introduced by pdsolve. This format enables one to see how particular the solution obtained is. In these cases, an explicit result for the indeterminate function, a particular solution, can be obtained by calling pdsolve with the extra argument build, or from this PDE solution structure by using the PDEtools[build] command. See the Examples section.

c) When pdsolve fails, it returns NULL.

You can change this scheme for returning solutions of pdsolve to only return a solution when it is the most general one, item a) above, and in all other cases return NULL, by assigning _Env_pdsolve_generalsolution := true.

You can change the default ways of expressing results by assigning the values $1$ or $2$ to the environment variable _EnvBuildPdsolve. By default, _EnvBuildPdsolve is assigned to $1$. When _EnvBuildPdsolve is set to $2$, pdsolve always tries to build an explicit solution, independent of the generality of the result obtained.

The arbitrary functions appearing in the output of pdsolve are of the form $f_n$, with $n$ an integer. These are literal subscripts symbols constructed, e.g., as in f__1, displayed as $f_1$. Optionally, these functions can be displayed $\mathrm{\_Fn}$ as in Maple releases prior to 2023; for that purpose call pdsolve once with the extra argument arbitraryfunctions = traditional, and all subsequent calls will automatically display the arbitrary functions using the traditional form $\mathrm{\_Fn}$. You can also indicate the traditional form to be the default display by adding this line to your maple initialization file: `pdsolve/arbitraryfunctions` = traditional. After that, to return to the default display $f_n$ it suffices to do one call to pdsolve with the optional argument arbitraryfunctions = subscripted. NOTE: when displaying these functions as $f_n$, an alias $f_n=\mathrm{\_Fn}$ is automatically set, so that programs written expecting the traditional display $\mathrm{\_Fn}$ of previous releases work the same way, without requiring any changes.

$\mathrm{PDE}≔x\mathrm{diff}\left(f\left(x\,y\right)\,y\right)-\mathrm{diff}\left(f\left(x\,y\right)\,x\right)=\frac{{f\left(x\,y\right)}^{2}g\left(x\right)}{h\left(y\right)}$

$\mathrm{PDE}≔x\left(\frac{\partial }{\partial y}f\left(x\,y\right)\right)-\frac{\partial }{\partial x}f\left(x\,y\right)=\frac{{f\left(x\,y\right)}^{2}g\left(x\right)}{h\left(y\right)}$

$\mathrm{ans}≔\mathrm{pdsolve}\left(\mathrm{PDE}\right)$

$\mathrm{ans}≔f\left(x\,y\right)=\frac{1}{{\int }_{\phantom{\mathrm{`\; `}}}^x\frac{g\left(\mathrm{\_a}\right)}{h\left(-\frac{{\mathrm{\_a}}^2}{2}+\frac{x^2}{2}+y\right)}\ⅆ\mathrm{\_a}+\mathrm{f\_\_1}\left(\frac{x^2}{2}+y\right)}$

$\mathrm{pdetest}\left(\mathrm{ans}\,\mathrm{PDE}\right)$

$0$

$\mathrm{PDE}≔\mathrm{Diff}\left(r^2\mathrm{diff}\left(F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\,r\right)\,r\right)+\frac{1}{\sin \left(\mathrm{\theta }\right)}\mathrm{Diff}\left(\sin \left(\mathrm{\theta }\right)\mathrm{diff}\left(F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\,\mathrm{\theta }\right)\,\mathrm{\theta }\right)+\frac{1}{{\sin \left(\mathrm{\theta }\right)}^2}\mathrm{diff}\left(F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\,\mathrm{\phi }\,\mathrm{\phi }\right)=0$

$\mathrm{PDE}≔\frac{\partial }{\partial r}\left(r^2\left(\frac{\partial }{\partial r}F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)\right)+\frac{\frac{\partial }{\partial \mathrm{\theta }}\left(\sin \left(\mathrm{\theta }\right)\left(\frac{\partial }{\partial \mathrm{\theta }}F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)\right)}{\sin \left(\mathrm{\theta }\right)}+\frac{\frac{{\partial }^2}{\partial {\mathrm{\phi }}^2}F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)}{{\sin \left(\mathrm{\theta }\right)}^2}=0$

$\mathrm{ans}≔\mathrm{pdsolve}\left(\mathrm{PDE}\right)$

$\mathrm{ans}≔F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)=\mathrm{f\_\_1}\left(r\right)\mathrm{f\_\_2}\left(\mathrm{\theta }\right)\mathrm{f\_\_3}\left(\mathrm{\phi }\right)\mathbf{where}\left[\left\{\frac{{\ⅆ}^2}{\ⅆr^2}\mathrm{f\_\_1}\left(r\right)=\frac{\mathrm{f\_\_1}\left(r\right){\mathrm{\_c}}_1}{r^2}-\frac{2\left(\frac{\ⅆ}{\ⅆr}\mathrm{f\_\_1}\left(r\right)\right)}{r}\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\theta }}^2}\mathrm{f\_\_2}\left(\mathrm{\theta }\right)=-\mathrm{f\_\_2}\left(\mathrm{\theta }\right){\mathrm{\_c}}_1+\frac{\mathrm{f\_\_2}\left(\mathrm{\theta }\right){\mathrm{\_c}}_2}{{\sin \left(\mathrm{\theta }\right)}^2}-\frac{\cos \left(\mathrm{\theta }\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\theta }}\mathrm{f\_\_2}\left(\mathrm{\theta }\right)\right)}{\sin \left(\mathrm{\theta }\right)}\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\phi }}^2}\mathrm{f\_\_3}\left(\mathrm{\phi }\right)=-\mathrm{f\_\_3}\left(\mathrm{\phi }\right){\mathrm{\_c}}_2\right\}\right]$

$\mathrm{simplify}\left(\mathrm{pdsolve}\left(\mathrm{PDE}\,\mathrm{build}\right)\,\mathrm{size}\right)\mathbf{assuming}0\le \mathrm{\theta },\mathrm{\theta }\le \mathrm{\pi }$

$F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)=\frac{{\sin \left(\mathrm{\theta }\right)}^{\sqrt{{\mathrm{\_c}}_2}}{\left(\mathrm{-1}\right)}^{\frac{\sqrt{{\mathrm{\_c}}_2}}{2}}\left(\mathrm{c\_\_1}{r}^{\frac{\sqrt{1+4{\mathrm{\_c}}_1}}{2}}+\mathrm{c\_\_2}{r}^{-\frac{\sqrt{1+4{\mathrm{\_c}}_1}}{2}}\right)\left(\mathrm{c\_\_5}\sin \left(\sqrt{{\mathrm{\_c}}_2}\mathrm{\phi }\right)+\mathrm{c\_\_6}\cos \left(\sqrt{{\mathrm{\_c}}_2}\mathrm{\phi }\right)\right)\left(\cos \left(\mathrm{\theta }\right)\mathrm{hypergeom}\left(\left[\frac{\sqrt{{\mathrm{\_c}}_2}}{2}+\frac{\sqrt{1+4{\mathrm{\_c}}_1}}{4}+\frac{3}{4}\,\frac{\sqrt{{\mathrm{\_c}}_2}}{2}-\frac{\sqrt{1+4{\mathrm{\_c}}_1}}{4}+\frac{3}{4}\right]\,\left[\frac{3}{2}\right]\,\frac{\cos \left(2\mathrm{\theta }\right)}{2}+\frac{1}{2}\right)\mathrm{c\_\_4}+\mathrm{c\_\_3}\mathrm{hypergeom}\left(\left[\frac{\sqrt{{\mathrm{\_c}}_2}}{2}+\frac{\sqrt{1+4{\mathrm{\_c}}_1}}{4}+\frac{1}{4}\,\frac{\sqrt{{\mathrm{\_c}}_2}}{2}-\frac{\sqrt{1+4{\mathrm{\_c}}_1}}{4}+\frac{1}{4}\right]\,\left[\frac{1}{2}\right]\,\frac{\cos \left(2\mathrm{\theta }\right)}{2}+\frac{1}{2}\right)\right)}{\sqrt{r}}$

$\mathrm{PDE}≔-\mathrm{diff}\left(S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)\,t\right)=\frac{\frac{1}{2}{\mathrm{diff}\left(S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)\,\mathrm{\xi }\right)}^2\left({\mathrm{\xi }}^2-1\right)}{{\mathrm{\sigma }}^{2}m\left({\mathrm{\xi }}^2-{\mathrm{\eta }}^2\right)}+\frac{\frac{1}{2}{\mathrm{diff}\left(S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)\,\mathrm{\eta }\right)}^2\left(1-{\mathrm{\eta }}^2\right)}{{\mathrm{\sigma }}^{2}m\left({\mathrm{\xi }}^2-{\mathrm{\eta }}^2\right)}+\frac{\frac{1}{2}{\mathrm{diff}\left(S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)\,\mathrm{\phi }\right)}^2}{m{\mathrm{\sigma }}^2\left({\mathrm{\xi }}^2-1\right)\left(1-{\mathrm{\eta }}^2\right)}+\frac{a\left(\mathrm{\xi }\right)+b\left(\mathrm{\eta }\right)}{{\mathrm{\xi }}^2-{\mathrm{\eta }}^2}$

$\mathrm{PDE}≔-\frac{\partial }{\partial t}S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)=\frac{{\left(\frac{\partial }{\partial \mathrm{\xi }}S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)\right)}^2\left({\mathrm{\xi }}^2-1\right)}{2{\mathrm{\sigma }}^{2}m\left(-{\mathrm{\eta }}^2+{\mathrm{\xi }}^2\right)}+\frac{{\left(\frac{\partial }{\partial \mathrm{\eta }}S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)\right)}^2\left(-{\mathrm{\eta }}^2+1\right)}{2m{\mathrm{\sigma }}^2\left(-{\mathrm{\eta }}^2+{\mathrm{\xi }}^2\right)}+\frac{{\left(\frac{\partial }{\partial \mathrm{\phi }}S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)\right)}^2}{2m{\mathrm{\sigma }}^2\left({\mathrm{\xi }}^2-1\right)\left(-{\mathrm{\eta }}^2+1\right)}+\frac{a\left(\mathrm{\xi }\right)+b\left(\mathrm{\eta }\right)}{-{\mathrm{\eta }}^2+{\mathrm{\xi }}^2}$

$\mathrm{ans}≔\mathrm{pdsolve}\left(\mathrm{PDE}\right)$

$\mathrm{ans}≔S\left(t\,\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\phi }\right)=\mathrm{f\_\_1}\left(t\right)+\mathrm{f\_\_2}\left(\mathrm{\xi }\right)+\mathrm{f\_\_3}\left(\mathrm{\eta }\right)+\mathrm{f\_\_4}\left(\mathrm{\phi }\right)\mathbf{where}\left[\left\{{\left(\frac{\ⅆ}{\ⅆ\mathrm{\xi }}\mathrm{f\_\_2}\left(\mathrm{\xi }\right)\right)}^2=-\frac{2m{\mathrm{\sigma }}^2{\mathrm{\xi }}^2{\mathrm{\_c}}_1}{{\mathrm{\xi }}^2-1}-\frac{2{\mathrm{\sigma }}^{2}m{\mathrm{\_c}}_3}{{\mathrm{\xi }}^2-1}-\frac{{\mathrm{\_c}}_4^2}{{\mathrm{\xi }}^4-2{\mathrm{\xi }}^2+1}-\frac{2a\left(\mathrm{\xi }\right){\mathrm{\sigma }}^{2}m}{{\mathrm{\xi }}^2-1}\,{\left(\frac{\ⅆ}{\ⅆ\mathrm{\eta }}\mathrm{f\_\_3}\left(\mathrm{\eta }\right)\right)}^2=-\frac{2{\mathrm{\eta }}^{2}m{\mathrm{\sigma }}^2{\mathrm{\_c}}_1}{{\mathrm{\eta }}^2-1}-\frac{2{\mathrm{\sigma }}^{2}m{\mathrm{\_c}}_3}{{\mathrm{\eta }}^2-1}-\frac{{\mathrm{\_c}}_4^2}{{\left({\mathrm{\eta }}^2-1\right)}^2}+\frac{2b\left(\mathrm{\eta }\right){\mathrm{\sigma }}^{2}m}{{\mathrm{\eta }}^2-1}\,\frac{\ⅆ}{\ⅆt}\mathrm{f\_\_1}\left(t\right)={\mathrm{\_c}}_1\,\frac{\ⅆ}{\ⅆ\mathrm{\phi }}\mathrm{f\_\_4}\left(\mathrm{\phi }\right)={\mathrm{\_c}}_4\right\}\right]$

$\mathrm{pdetest}\left(\mathrm{ans}\,\mathrm{PDE}\right)$

$0$

$\mathrm{PDE}≔S\left(x\,y\right)\mathrm{diff}\left(S\left(x\,y\right)\,y\,x\right)+\mathrm{diff}\left(S\left(x\,y\right)\,x\right)\mathrm{diff}\left(S\left(x\,y\right)\,y\right)=1$

$\mathrm{PDE}≔S\left(x\,y\right)\left(\frac{{\partial }^2}{\partial x\partial y}S\left(x\,y\right)\right)+\left(\frac{\partial }{\partial x}S\left(x\,y\right)\right)\left(\frac{\partial }{\partial y}S\left(x\,y\right)\right)=1$

$\mathrm{infolevel}\left[\mathrm{pdsolve}\right]≔3\:$

$\mathrm{pdsolve}\left(\mathrm{PDE}\,\mathrm{generalsolution}\right)$

First set of solution methods (general or quasi general solution)
   -> trying missing independent variables
   -> trying the Laplace's method
   -> trying a first integral mapping the PDE into and ODE plus parameters
   <- first integral method PDE into an ODE plus parameters successful.
<- First set of solution methods successful
<- Returning a *general* solution

$S\left(x\&comma;y\right)=\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\mathrm{f\_\_1}\left(y\right)-2xy-\mathrm{f\_\_2}\left(x\right)\right)$

$\mathrm{infolevel}\left[\mathrm{pdsolve}\right]≔1\&colon;$

$\mathrm{struc}≔\mathrm{pdsolve}\left(\mathrm{PDE}\&comma;\mathrm{HINT}=f\left(x\right)g\left(y\right)\right)$

$\mathrm{struc}≔S\left(x\&comma;y\right)=f\left(x\right)g\left(y\right)\mathbf{where}\left[\left\{\frac{\&DifferentialD;}{\&DifferentialD;x}f\left(x\right)=\frac{{\mathrm{\_c}}_1}{f\left(x\right)}\&comma;\frac{\&DifferentialD;}{\&DifferentialD;y}g\left(y\right)=\frac{1}{2g\left(y\right){\mathrm{\_c}}_1}\right\}\right]$

$\mathrm{PDEtools}\left[\mathrm{build}\right]\left(\mathrm{struc}\right)$

$S\left(x\&comma;y\right)=\frac{\sqrt{2x{\mathrm{\_c}}_1+\mathrm{c\_\_1}}\sqrt{\mathrm{c\_\_2}{\mathrm{\_c}}_1^2+{\mathrm{\_c}}_{1}y}}{{\mathrm{\_c}}_1}$

$\mathrm{pdsolve}\left(\mathrm{PDE}\&comma;\mathrm{HINT}={P\left(x\&comma;y\right)}^{\frac{1}{2}}\right)$

$S\left(x\&comma;y\right)=\sqrt{\mathrm{f\_\_2}\left(x\right)+\mathrm{f\_\_1}\left(y\right)+2xy}$

$\mathrm{PDE}≔\mathrm{diff}\left(f\left(x\&comma;y\&comma;z\right)\&comma;x\right)+{\mathrm{diff}\left(f\left(x\&comma;y\&comma;z\right)\&comma;y\right)}^2=f\left(x\&comma;y\&comma;z\right)+z$

$\mathrm{PDE}≔\frac{\partial }{\partial x}f\left(x\&comma;y\&comma;z\right)+{\left(\frac{\partial }{\partial y}f\left(x\&comma;y\&comma;z\right)\right)}^2=f\left(x\&comma;y\&comma;z\right)+z$

$\mathrm{pdsolve}\left(\mathrm{PDE}\&comma;\mathrm{HINT}=\mathrm{strip}\right)$

$\frac{\partial }{\partial x}f\left(x\&comma;y\&comma;z\right)+{\left(\frac{\partial }{\partial y}f\left(x\&comma;y\&comma;z\right)\right)}^2-f\left(x\&comma;y\&comma;z\right)-z=0\mathbf{where}\left[\left\{\left\{f\left(\mathrm{\_s}\right)=\mathrm{c\_\_1}{\&ExponentialE;}^{2\mathrm{\_s}}+\mathrm{c\_\_4}{\&ExponentialE;}^{\mathrm{\_s}}-\mathrm{c\_\_5}\&comma;x\left(\mathrm{\_s}\right)=\mathrm{\_s}+\mathrm{c\_\_6}\&comma;y\left(\mathrm{\_s}\right)=2\mathrm{c\_\_3}{\&ExponentialE;}^{\mathrm{\_s}}+\mathrm{c\_\_2}\&comma;z\left(\mathrm{\_s}\right)=\mathrm{c\_\_5}\&comma;{\mathrm{\_p}}_1\left(\mathrm{\_s}\right)=\mathrm{c\_\_4}{\&ExponentialE;}^{\mathrm{\_s}}\&comma;{\mathrm{\_p}}_2\left(\mathrm{\_s}\right)=\mathrm{c\_\_3}{\&ExponentialE;}^{\mathrm{\_s}}\right\}\right\}\&comma;\phantom{}\mathbf{and}\left\{{\mathrm{\_p}}_1=\frac{\partial }{\partial x}f\left(x\&comma;y\&comma;z\right)\&comma;{\mathrm{\_p}}_2=\frac{\partial }{\partial y}f\left(x\&comma;y\&comma;z\right)\right\}\right]$

$\mathrm{with}\left(\mathrm{Physics}\right)$

$\left[\mathrm{`*`}\&comma;\mathrm{`.`}\&comma;\mathrm{Annihilation}\&comma;\mathrm{AntiCommutator}\&comma;\mathrm{Antisymmetrize}\&comma;\mathrm{Assume}\&comma;\mathrm{Bra}\&comma;\mathrm{Bracket}\&comma;\mathrm{Check}\&comma;\mathrm{Christoffel}\&comma;\mathrm{Coefficients}\&comma;\mathrm{Commutator}\&comma;\mathrm{CompactDisplay}\&comma;\mathrm{Coordinates}\&comma;\mathrm{Creation}\&comma;\mathrm{D\_}\&comma;\mathrm{Dagger}\&comma;\mathrm{Decompose}\&comma;\mathrm{Define}\&comma;\mathrm{D\&gamma;}\&comma;\mathrm{DiracConjugate}\&comma;\mathrm{Einstein}\&comma;\mathrm{EnergyMomentum}\&comma;\mathrm{Expand}\&comma;\mathrm{ExteriorDerivative}\&comma;\mathrm{Factor}\&comma;\mathrm{FeynmanDiagrams}\&comma;\mathrm{FeynmanIntegral}\&comma;\mathrm{Fundiff}\&comma;\mathrm{Geodesics}\&comma;\mathrm{GrassmannParity}\&comma;\mathrm{Gtaylor}\&comma;\mathrm{Intc}\&comma;\mathrm{Inverse}\&comma;\mathrm{Ket}\&comma;\mathrm{KillingVectors}\&comma;\mathrm{KroneckerDelta}\&comma;\mathrm{LagrangeEquations}\&comma;\mathrm{LeviCivita}\&comma;\mathrm{Library}\&comma;\mathrm{LieBracket}\&comma;\mathrm{LieDerivative}\&comma;\mathrm{Normal}\&comma;\mathrm{NumericalRelativity}\&comma;\mathrm{Parameters}\&comma;\mathrm{PerformOnAnticommutativeSystem}\&comma;\mathrm{Projector}\&comma;\mathrm{Psigma}\&comma;\mathrm{Redefine}\&comma;\mathrm{Ricci}\&comma;\mathrm{Riemann}\&comma;\mathrm{Setup}\&comma;\mathrm{Simplify}\&comma;\mathrm{SortProducts}\&comma;\mathrm{SpaceTimeVector}\&comma;\mathrm{StandardModel}\&comma;\mathrm{Substitute}\&comma;\mathrm{SubstituteTensor}\&comma;\mathrm{SubstituteTensorIndices}\&comma;\mathrm{SumOverRepeatedIndices}\&comma;\mathrm{Symmetrize}\&comma;\mathrm{TensorArray}\&comma;\mathrm{Tetrads}\&comma;\mathrm{ThreePlusOne}\&comma;\mathrm{ToContravariant}\&comma;\mathrm{ToCovariant}\&comma;\mathrm{ToFieldComponents}\&comma;\mathrm{ToSuperfields}\&comma;\mathrm{Trace}\&comma;\mathrm{TransformCoordinates}\&comma;\mathrm{Vectors}\&comma;\mathrm{Weyl}\&comma;\mathrm{`^`}\&comma;\mathrm{dAlembertian}\&comma;\mathrm{d\_}\&comma;\mathrm{diff}\&comma;\mathrm{g\_}\&comma;\mathrm{gamma\_}\right]$

$\mathrm{Setup}\left(\mathrm{anticommutativepre}=\left\{Q\&comma;\mathrm{\theta }\right\}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{Q\&comma;\mathrm{\theta }\right\}\right]$

$\mathrm{diff}\left(Q\left(x\&comma;y\&comma;\mathrm{\theta }\right)\&comma;x\&comma;\mathrm{\theta }\right)=0$

$\frac{{\partial }^2}{\partial x\partial \mathrm{\theta }}Q\left(x\&comma;y\&comma;\mathrm{\theta }\right)=0$

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

$Q\left(x\&comma;y\&comma;\mathrm{\theta }\right)=\mathrm{f\_\_1}\left(x\&comma;y\right)\mathrm{\_\&lambda;1}+\mathrm{f\_\_3}\left(y\right)\mathrm{\theta }$

$\mathrm{Physics}:-\mathrm{GrassmannParity}\left(\right)$

$1=1$

Cheb-Terrab, E.S., and von Bulow, K. "A Computational Approach for the Analytical Solving of Partial Differential Equations". Computer Physics Communications. Vol. 90. (1995): 102-116.

The generalsolution option was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.

The pdsolve command was updated in Maple 2023.

The arbitraryfunctions option was introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.