The LHSolve method attempts to solve the linear homogeneous PDEs in a LHPDE object.

If solving is successful then the method returns a list of equations as the general solution.

By specifying output = "basis", the returned output will be a list of lists of equations representing each solution in a basis.

By specifying output = "lhpde", the returned output will be a new LHPDE object that is fully integrated.

For a returned LHPDE object S that involves constants of integration variables, these variables are treated as additional dependent variables of S. The default names are _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 ${\mathrm{\alpha }}_1\,{\mathrm{\alpha }}_2\,{\mathrm{\alpha }}_3\,\dots$

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 existing pdsolve command (for partial DEs system) and the dsolve command (for ordinary DEs system) of finite type.

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

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

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

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

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

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

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

$\mathrm{IsFiniteType}\left(S\right)$

$\mathrm{true}$

$\mathrm{LHSolve}\left(S\right)$

$\left[\mathrm{\xi }=-\mathrm{c\_\_1}y+\mathrm{c\_\_3}\,\mathrm{\eta }=\mathrm{c\_\_1}x+\mathrm{c\_\_2}\right]$

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

$\left[\mathrm{\xi }=-\mathrm{\alpha }y+\mathrm{\delta }\,\mathrm{\eta }=\mathrm{\alpha }x+\mathrm{\beta }\right]$

$\mathrm{LHSolve}\left(S\,\mathrm{output}=''basis''\right)$

$\left[\left[\mathrm{\xi }=-y\,\mathrm{\eta }=x\right]\,\left[\mathrm{\xi }=0\,\mathrm{\eta }=1\right]\,\left[\mathrm{\xi }=1\,\mathrm{\eta }=0\right]\right]$

$\mathrm{LHSolve}\left(S\,\mathrm{output}=''lhpde''\,\mathrm{consts}=\mathrm{\alpha }\right)$

$\left[\mathrm{\xi }=-y{\mathrm{\alpha }}_1+{\mathrm{\alpha }}_3\,\mathrm{\eta }=x{\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\,{\mathrm{\alpha }}_1\,{\mathrm{\alpha }}_2\,{\mathrm{\alpha }}_3\right]$

The LHSolve command was introduced in Maple 2020.

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