ToMissingDependentVariable receives a a partial differential equation (PDE), typically depending explicitly on the dependent variable U - say $u\left(x\,y\,\mathrm{...}\right)$, where the independent variables are $\left(x\,y\,\mathrm{...}\right)\=X$, and returns another PDE for a a new dependent variable $v\left(X\,u\right)$, that depend on $v\left(X\,u\right)$ only through its derivatives with respect to $X,u$. The output actually consists of a sequence of two objects, the first being the PDE in $v\left(X\,u\right)$, the second being $v\left(X\,u\right)$ itself.

The relevance of this command is in that from the knowledge of the solution of the PDE for $v\left(X\,u\right)$ one can write, directly, the solution to the original PDE for $u\left(X\right)$, as shown below in the Examples section.

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

$x\left({\left(\frac{\partial }{\partial x}m\left(x\,y\right)\right)}^3\+{\left(\frac{\partial }{\partial y}m\left(x\,y\right)\right)}^3\right)\=m\left(x\,y\right)\left(\frac{\partial }{\partial x}m\left(x\,y\right)\right)$

$x\left({\left(\frac{\partial }{\partial x}m\left(x\,y\right)\right)}^3+{\left(\frac{\partial }{\partial y}m\left(x\,y\right)\right)}^3\right)=m\left(x\,y\right)\left(\frac{\partial }{\partial x}m\left(x\,y\right)\right)$

$\mathrm{ToMissingDependentVariable}\left(\,m\left(x\,y\right)\,v\right)$

$\frac{x\left(-{\left(\frac{\partial }{\partial x}v\left(x\,y\,m\right)\right)}^3-{\left(\frac{\partial }{\partial y}v\left(x\,y\,m\right)\right)}^3\right)}{{\left(\frac{\partial }{\partial m}v\left(x\,y\,m\right)\right)}^3}=-\frac{m\left(\frac{\partial }{\partial x}v\left(x\,y\,m\right)\right)}{\frac{\partial }{\partial m}v\left(x\,y\,m\right)},v\left(x\,y\,m\right)$

$\mathrm{pdsolve}\left(\,\mathrm{build}\right)$

$v\left(x\,y\,m\right)=\frac{{12}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\int \frac{{\left(-9x^3{\mathrm{\_c}}_2^3+x^2\sqrt{3}\sqrt{27x^2{\mathrm{\_c}}_2^6+\frac{4{\mathrm{\_c}}_1^3}{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{x}\ⅆx\right)}{6}-\frac{{\mathrm{\_c}}_1{12}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\int \frac{1}{{\left(-9x^3{\mathrm{\_c}}_2^3+x^2\sqrt{3}\sqrt{27x^2{\mathrm{\_c}}_2^6+\frac{4{\mathrm{\_c}}_1^3}{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\ⅆx\right)}{6}+\mathrm{c\_\_1}+{\mathrm{\_c}}_{2}y+\mathrm{c\_\_2}-2\sqrt{-m{\mathrm{\_c}}_1}+\mathrm{c\_\_3}$

$\mathrm{\α}\=\mathrm{subs}\left(m\=m\left(x\,y\right)\,\mathrm{rhs}\left(\right)\right)$

$\mathrm{\alpha }=\frac{{12}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\int \frac{{\left(-9x^3{\mathrm{\_c}}_2^3+x^2\sqrt{3}\sqrt{27x^2{\mathrm{\_c}}_2^6+\frac{4{\mathrm{\_c}}_1^3}{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{x}\ⅆx\right)}{6}-\frac{{\mathrm{\_c}}_1{12}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\int \frac{1}{{\left(-9x^3{\mathrm{\_c}}_2^3+x^2\sqrt{3}\sqrt{27x^2{\mathrm{\_c}}_2^6+\frac{4{\mathrm{\_c}}_1^3}{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\ⅆx\right)}{6}+\mathrm{c\_\_1}+{\mathrm{\_c}}_{2}y+\mathrm{c\_\_2}-2\sqrt{-m\left(x\,y\right){\mathrm{\_c}}_1}+\mathrm{c\_\_3}$

$\mathrm{pdetest}\left(\,\right)$

$0$

$\mathrm{isolate}\left(\,m\left(x\,y\right)\right)$

$m\left(x\,y\right)=-\frac{{\left(-\frac{\mathrm{\alpha }}{2}+\frac{{12}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\int \frac{{\left(-9x^3{\mathrm{\_c}}_2^3+x^2\sqrt{3}\sqrt{27x^2{\mathrm{\_c}}_2^6+\frac{4{\mathrm{\_c}}_1^3}{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{x}\ⅆx\right)}{12}-\frac{{\mathrm{\_c}}_1{12}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\int \frac{1}{{\left(-9x^3{\mathrm{\_c}}_2^3+x^2\sqrt{3}\sqrt{27x^2{\mathrm{\_c}}_2^6+\frac{4{\mathrm{\_c}}_1^3}{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\ⅆx\right)}{12}+\frac{\mathrm{c\_\_1}}{2}+\frac{{\mathrm{\_c}}_{2}y}{2}+\frac{\mathrm{c\_\_2}}{2}+\frac{\mathrm{c\_\_3}}{2}\right)}^2}{{\mathrm{\_c}}_1}$

$\mathrm{pdetest}\left(\,\right)$

$0$

The PDEtools[ToMissingDependentVariable] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.