The remove_RootOf command receives an algebraic expression containing a RootOf and removes it by rewriting the expression. In general, this command is used to directly and naturally convey the mathematical meaning of algebraic expressions that contain RootOfs.

Note that remove_RootOf is different from allvalues; remove_RootOf attempts to remove the RootOf -- typically resulting in implicit algebraic expressions -- while allvalues attempts to return all the values implied by the RootOf.

remove_RootOf is of particular value for understanding or further manipulating ODE solutions, and especially when allvalues fails in evaluating the RootOf.

When the given expression has no RootOf inside it, remove_RootOf returns it as given; when the expression has many RootOfs, only one RootOf (the one with greatest length) is removed. It is also possible to direct remove_RootOf to remove a specific RootOf, say R, by passing R to remove_RootOf as its second argument.

When the removal of a RootOf itself requires the introduction of another RootOf, the removal is not performed and the expression received is returned as given. Also remove_RootOf only removes RootOfs of one argument (so RootOfs with indices or labels will remain in the output).

This function is part of the DEtools package, and so it can be used in the form remove_RootOf(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[remove_RootOf](..).

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

$\mathrm{diff}\left(y\left(x\right)\,x\right)=-\frac{1}{2}\left(a{x}^2-2F\left(y\left(x\right)+\frac{1}{8}a{x}^4\right)\right)x$

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=-\frac{\left(a{x}^2-2F\left(y\left(x\right)+\frac{a{x}^4}{8}\right)\right)x}{2}$

$\mathrm{dsolve}\left(\right)$

$y\left(x\right)=-\frac{a{x}^4}{8}+\mathrm{RootOf}\left(-x^2+2\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{F\left(\mathrm{\_a}\right)}\ⅆ\mathrm{\_a}\right)+2\mathrm{c\_\_1}\right)$

$\mathrm{remove\_RootOf}\left(\right)$

$-x^2+2\left({\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)+\frac{a{x}^4}{8}}\frac{1}{F\left(\mathrm{\_a}\right)}\ⅆ\mathrm{\_a}\right)+2\mathrm{c\_\_1}=0$

$\mathrm{expr}≔\frac{y}{x}+\mathrm{RootOf}\left(\mathrm{\_Z}+\exp \left(\mathrm{\_Z}\right)\right)=C$

$\mathrm{expr}≔\frac{y}{x}+\mathrm{RootOf}\left(\mathrm{\_Z}+{\ⅇ}^{\mathrm{\_Z}}\right)=C$

$\mathrm{DEtools}\left[\mathrm{remove\_RootOf}\right]\left(\mathrm{expr}\right)$

$-\frac{y}{x}+C+{\ⅇ}^{-\frac{y}{x}+C}=0$

$\mathrm{allvalues}\left(\mathrm{expr}\right)$

$\frac{y}{x}-\mathrm{LambertW}\left(\mathrm{\_Z2~}\,1\right)=C$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=\frac{\left(xy\left(x\right)+1\right)\left(x^2{y\left(x\right)}^2+x^{2}y\left(x\right)+2xy\left(x\right)+1+x+x^2\right)}{x^5}$

$\mathrm{ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\frac{\left(xy\left(x\right)+1\right)\left(x^2{y\left(x\right)}^2+x^{2}y\left(x\right)+2xy\left(x\right)+1+x+x^2\right)}{x^5}$

$\mathrm{dsolve}\left(\mathrm{ode}\right)$

$y\left(x\right)=\frac{17\mathrm{RootOf}\left(162\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{289{\mathrm{\_a}}^3+54\mathrm{\_a}-54}\ⅆ\mathrm{\_a}\right)x+3\mathrm{c\_\_1}x+2\right)x-3x-9}{9x}$

$\mathrm{sol}≔\mathrm{value}\left(\right)$

$\mathrm{sol}≔y\left(x\right)=\frac{17\mathrm{RootOf}\left(54\left(\sum _{\mathrm{\_R}=\mathrm{RootOf}\left(289{\mathrm{\_Z}}^3+54\mathrm{\_Z}-54\right)}\frac{\ln \left(\mathrm{\_Z}-\mathrm{\_R}\right)}{289{\mathrm{\_R}}^2+18}\right)x+3\mathrm{c\_\_1}x+2\right)x-3x-9}{9x}$

$\mathrm{remove\_RootOf}\left(\mathrm{sol}\right)$

$54\left(\sum _{\mathrm{\_R}=\mathrm{RootOf}\left(289{\mathrm{\_Z}}^3+54\mathrm{\_Z}-54\right)}\frac{\ln \left(\frac{3\left(3xy\left(x\right)+x+3\right)}{17x}-\mathrm{\_R}\right)}{289{\mathrm{\_R}}^2+18}\right)x+3\mathrm{c\_\_1}x+2=0$

$\mathrm{odetest}\left(\,\mathrm{ode}\right)$

$0$

$\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-\frac{h\left(\frac{y\left(x\right)}{\mathrm{sqrt}\left(x\right)}\right)}{x^{\frac{3}{2}}}=0$

$\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-\frac{h\left(\frac{y\left(x\right)}{\sqrt{x}}\right)}{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=0$

$\left[\mathrm{dsolve}\left(\right)\right]$

$\left[y\left(x\right)=\mathrm{RootOf}\left(\mathrm{\_Z}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+4h\left(\frac{\mathrm{\_Z}}{\sqrt{x}}\right)x^2\right)\,y\left(x\right)=\mathrm{RootOf}\left(-\ln \left(x\right)+2\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{\sqrt{\mathrm{c\_\_1}+8\left(\int h\left(\mathrm{\_g}\right)\ⅆ\mathrm{\_g}\right)+{\mathrm{\_g}}^2}}\ⅆ\mathrm{\_g}\right)+2\mathrm{c\_\_2}\right)\sqrt{x}\,y\left(x\right)=\mathrm{RootOf}\left(-\ln \left(x\right)-2\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{\sqrt{\mathrm{c\_\_1}+8\left(\int h\left(\mathrm{\_g}\right)\ⅆ\mathrm{\_g}\right)+{\mathrm{\_g}}^2}}\ⅆ\mathrm{\_g}\right)+2\mathrm{c\_\_2}\right)\sqrt{x}\right]$

$\mathrm{map}\left(\mathrm{remove\_RootOf}\,\right)$

$\left[y\left(x\right)x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+4h\left(\frac{y\left(x\right)}{\sqrt{x}}\right)x^2=0\,-\ln \left(x\right)+2\left({\int }_{\phantom{\mathrm{`\; `}}}^{\frac{y\left(x\right)}{\sqrt{x}}}\frac{1}{\sqrt{\mathrm{c\_\_1}+8\left(\int h\left(\mathrm{\_g}\right)\ⅆ\mathrm{\_g}\right)+{\mathrm{\_g}}^2}}\ⅆ\mathrm{\_g}\right)+2\mathrm{c\_\_2}=0\,-\ln \left(x\right)-2\left({\int }_{\phantom{\mathrm{`\; `}}}^{\frac{y\left(x\right)}{\sqrt{x}}}\frac{1}{\sqrt{\mathrm{c\_\_1}+8\left(\int h\left(\mathrm{\_g}\right)\ⅆ\mathrm{\_g}\right)+{\mathrm{\_g}}^2}}\ⅆ\mathrm{\_g}\right)+2\mathrm{c\_\_2}=0\right]$