The AreSame method checks if the two LAVF objects obj1 and obj2 are the same.

This method is front-end to the AreSame method of a LHPDE object. That is, let S1, S2 be the determining systems of LAVF objects L1, L2 respectively, then AreSame(L1, L2) equals AreSame(S1,S2).

All optional arguments in this method will pass down to the AreSame method of a LHPDE object. See AreSame of a LHPDE object for how to apply optional arguments.

In the second calling sequence, the word criterion is provided as alias for criteria.

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

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

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

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

$\mathrm{Vp}≔\mathrm{VectorField}\left(\mathrm{\alpha }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\beta }\left(x\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$\mathrm{Vp}≔\mathrm{\alpha }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\beta }\left(x\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

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

$\mathrm{E2}≔\left[\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)=-\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]$

$\mathrm{E2p}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\alpha }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\beta }\left(x\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\alpha }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\alpha }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\alpha }\,\mathrm{\beta }\right]\right)$

$\mathrm{E2p}≔\left[\frac{{\partial }^2}{\partial y^2}\mathrm{\alpha }\left(x\,y\right)=0\,\frac{\ⅆ}{\ⅆx}\mathrm{\beta }\left(x\right)=-\frac{\partial }{\partial y}\mathrm{\alpha }\left(x\,y\right)\,\frac{\partial }{\partial x}\mathrm{\alpha }\left(x\,y\right)=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\alpha }\left(x\,y\right)\,\mathrm{\beta }\left(x\right)\right]$

$\mathrm{LE2}≔\mathrm{LAVF}\left(V\,\mathrm{E2}\right)$

$\mathrm{LE2}≔\left[\mathrm{\xi }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)=-\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)=0\right]\right\}$

$\mathrm{LE2p}≔\mathrm{LAVF}\left(\mathrm{Vp}\,\mathrm{E2p}\right)$

$\mathrm{LE2p}≔\left[\mathrm{\alpha }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\beta }\left(x\right)\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\frac{{\partial }^2}{\partial y^2}\mathrm{\alpha }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\alpha }\left(x\,y\right)=0\,\frac{\ⅆ}{\ⅆx}\mathrm{\beta }\left(x\right)=-\frac{\partial }{\partial y}\mathrm{\alpha }\left(x\,y\right)\right]\right\}$

$\mathrm{AreSame}\left(\mathrm{LE2}\,\mathrm{LE2p}\right)$

$\mathrm{true}$

$\mathrm{AreSame}\left(\mathrm{LE2}\,\mathrm{LE2p}\,\mathrm{criterion}=''sameSystem''\right)$

$\mathrm{false}$

$\mathrm{AreSame}\left(\mathrm{LE2}\,\mathrm{LE2p}\,\mathrm{criteria}=''sameSolutions''\right)$

$\mathrm{true}$

The AreSame command was introduced in Maple 2020.

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