The LAVFSolve method attempts to solve the determining system in a LAVF object.

If solving is successful then by default the method returns a list of solution vector fields.

The returned output can be as a basis (by specifying output = "basis") or a new LAVF object (by specifying output = "lavf").

For a returned output that involves constants of integration variables, by default these variables are labeled as  _C1, _C2, ...

The constant of integration variables can be renamed by specifying the optional argument consts = c.

By specifying consts = alpha (i.e. a name), the constants of integration will be named as ${\mathrm{\alpha }}_1\,{\mathrm{\alpha }}_2\,{\mathrm{\alpha }}_3\,..$

By specifying consts = [alpha, beta, phi...] (i.e. a list of names), the constants of integration will be named as $\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\phi }\,\dots$

This is a front-end to the LHSolve method for solving the determining system. LHSolve is associated with the LHPDE object, see Overview of the LHPDE object for more detail.

The method throws an exception if the LAVF is not of finite type.

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

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

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

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

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

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

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

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

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

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

$\mathrm{LAVFSolve}\left(L\right)$

$\left(-\mathrm{c\_\_1}y+\mathrm{c\_\_3}\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\left(\mathrm{c\_\_1}x+\mathrm{c\_\_2}\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{LAVFSolve}\left(L\,\mathrm{consts}=\left[\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\delta }\right]\right)$

$\left(-\mathrm{\alpha }y+\mathrm{\delta }\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\left(\mathrm{\alpha }x+\mathrm{\beta }\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{LAVFSolve}\left(L\,\mathrm{output}=''basis''\right)$

$\left[-y\left(\frac{\ⅆ}{\ⅆx}\right)+x\left(\frac{\ⅆ}{\ⅆy}\right)\,\frac{\ⅆ}{\ⅆy}\,\frac{\ⅆ}{\ⅆx}\right]$

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

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

The LAVFSolve command was introduced in Maple 2020.

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