Let L be a LAVF object that is either partial or fully integrated (i.e. its determining system includes constants or functions that are not infinitesimals). Then ImplicitForm(L) returns the implicit form of L, as a new LHPDE object.

The implicit form of L is defined by rif-reducing its determining system with respect to a block ranking $\mathrm{\xi }\ll a$ (i.e. all $\mathrm{\xi }$'s are ranked lower than any of $a$'s) where $\mathrm{\xi }\=\left({\mathrm{\xi }}^1\,\,{\mathrm{\xi }}^n\right)$ are infinitesimals and $a\=\left(a^1\,..\,a^t\right)$are non-infinitesimals such as constants of integration variables.

The returned output, a LHPDE object, is in rif-reduced form with ranking  $\mathrm{\xi }\ll a$ recorded. See Overview of the LHPDE object for more detail.

In the second calling sequence, the call returns a sub-system that includes infinitesimals only from the implicit form of L. This 'infinitesimals-only' sub-system is same as the non-integrated determining system of L.

If the input LAVF object is non-integrated (i.e. no constants of integration variables), then the implicit form of L is its determining system itself.

This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\:$

$V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$V≔\mathrm{\xi }\frac{\partial }{\partial x}+\mathrm{\eta }\frac{\partial }{\partial y}$

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$L≔\mathrm{LAVF}\left(V\,\mathrm{E2}\right)$

$L≔\left[\mathrm{\xi }\frac{\partial }{\partial x}+\mathrm{\eta }\frac{\partial }{\partial y}\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{Ls}≔\mathrm{LAVFSolve}\left(L\,\mathrm{output}=''lavf''\right)$

$\mathrm{Ls}≔\left[\mathrm{\xi }\frac{\partial }{\partial x}+\mathrm{\eta }\frac{\partial }{\partial y}\right]\&where\left\{\left[\mathrm{\xi }=-\mathrm{\_C1}y+\mathrm{\_C3}\,\mathrm{\eta }=\mathrm{\_C1}x+\mathrm{\_C2}\right]\right\}$

$\mathrm{Imp}≔\mathrm{ImplicitForm}\left(\mathrm{Ls}\right)$

$\mathrm{Imp}≔\left[\mathrm{\_C1}=-{\mathrm{\xi }}_y\,\mathrm{\_C2}=\left({\mathrm{\xi }}_y\right)x+\mathrm{\eta }\,\mathrm{\_C3}=-\left({\mathrm{\xi }}_y\right)y+\mathrm{\xi }\,{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{\_C1}\,\mathrm{\_C2}\,\mathrm{\_C3}\right]$

$\mathrm{type}\left(\mathrm{Imp}\,'\mathrm{LHPDE}'\right)$

$\mathrm{true}$

$\mathrm{GetRanking}\left(\mathrm{Imp}\right)$

$\left[\left[\mathrm{\_C1}\,\mathrm{\_C2}\,\mathrm{\_C3}\right]\,\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right]$

$S≔\mathrm{ImplicitForm}\left(\mathrm{Ls}\,\mathrm{infinitesimalsOnly}=\mathrm{true}\right)$

$S≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{AreSame}\left(S\,\mathrm{E2}\,\mathrm{criteria}=''sameSystem''\right)$

$\mathrm{true}$

The ImplicitForm command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.