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.

with(LieAlgebrasOfVectorFields):

Typesetting:-Settings(userep=true):

Typesetting:-Suppress([xi(x,y),eta(x,y)]):

V := VectorField(xi(x,y)*D[x] + eta(x,y)*D[y], space = [x,y]);

$V≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

E2 := LHPDE([diff(xi(x,y),y,y)=0, diff(eta(x,y),x)=-diff(xi(x,y),y), diff(eta(x,y),y)=0, diff(xi(x,y),x)=0], indep = [x,y], dep = [xi, eta]);

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

L := LAVF(V, E2);

$L≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

Ls := LAVFSolve(L, output= "lavf");

$\mathrm{Ls}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\mathrm{\xi }=-\mathrm{c\_\_1}y+\mathrm{c\_\_3}\,\mathrm{\eta }=\mathrm{c\_\_1}x+\mathrm{c\_\_2}\right]\right\}$

Imp := ImplicitForm(Ls);

$\mathrm{Imp}≔\left[\mathrm{c\_\_1}=-{\mathrm{\xi }}_y\,\mathrm{c\_\_2}=\left({\mathrm{\xi }}_y\right)x+\mathrm{\eta }\,\mathrm{c\_\_3}=-\left({\mathrm{\xi }}_y\right)y+\mathrm{\xi }\,{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\,\mathrm{c\_\_1}\,\mathrm{c\_\_2}\,\mathrm{c\_\_3}\right]$

type(Imp, 'LHPDE');

$\mathrm{true}$

GetRanking(Imp);

$\left[\left[\mathrm{c\_\_1}\,\mathrm{c\_\_2}\,\mathrm{c\_\_3}\right]\,\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right]$

S := ImplicitForm(Ls, infinitesimalsOnly = true);

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

AreSame(S, E2, criteria = "sameSystem");

$\mathrm{true}$

The ImplicitForm command was introduced in Maple 2020.

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