The routines translate and untranslate are used for the translation of DEs and operators.  In the case of a linear differential operator, the input des is given by a list (here A), which defines the operator by

The routine translate takes a DE or linear operator in the independent variable ivar, assumed to be centered about 0, and translates it to a DE or operator centered about pt.  Essentially, translate acts like DEtools[Dchangevar] for the restricted cases $t=\frac{1}{x}$ (pt = infinity) and $t=x+\mathrm{pt}$ (otherwise).

Likewise, the routine untranslate takes a DE, operator, or DE solution that is centered on the point pt and translates it to the equivalent centered on 0.

dvar must be provided in the instance that des is a differential equation or DE solution.  It is not required if des is an operator list.

Results may be "normalized" by way of DEtools[DEnormal].

Linear differential operators may be derived from DEs by way of DEtools[convertAlg].

These functions are part of the DEtools package, and so they can be used in the form translate(..) and untranslate(..) only after executing the command with(DEtools). However, they can always be accessed through the long form of the command DEtools[translate](..) or DEtools[untranslate](..).

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

$\mathrm{L\_A}≔\left[\frac{1}{x\left(x^5+2x+2\right)}\,x^2+1\,x^5+2x+2\right]\:$

$\mathrm{a1}≔\mathrm{translate}\left(\mathrm{L\_A}\,x\,\mathrm{\infty }\right)$

$\mathrm{a1}≔\left[\frac{x^6}{2x^5+2x^4+1}\,\frac{4x^5+3x^4-x^2+2}{x^2}\,\frac{2x^5+2x^4+1}{x}\right]$

$\mathrm{a2}≔\mathrm{untranslate}\left(\mathrm{a1}\,x\,\mathrm{\infty }\right)$

$\mathrm{a2}≔\left[\frac{1}{x\left(x^5+2x+2\right)}\,x^2+1\,x^5+2x+2\right]$

$\mathrm{DE}≔21\left(-2x^3+3x^2-5x+2\right)y\left(x\right)+42\left(x^2-x+1\right)x\left(x-1\right)\mathrm{D}\left(y\right)\left(x\right)+50x^3{\left(x-1\right)}^3{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(x\right)=x\sin \left(x\right)\:$

$\mathrm{de1}≔\mathrm{translate}\left(\mathrm{DE}\,x\,a\,y\left(x\right)\right)$

$\mathrm{de1}≔21\left(-2{\left(x+a\right)}^3+3{\left(x+a\right)}^2-5x-5a+2\right)y\left(x\right)+42\left({\left(x+a\right)}^2-x-a+1\right)\left(x+a\right)\left(x+a-1\right)\mathrm{D}\left(y\right)\left(x\right)+50{\left(x+a\right)}^3{\left(x+a-1\right)}^3{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(x\right)=\left(x+a\right)\sin \left(x+a\right)$

$\mathrm{de2}≔\mathrm{untranslate}\left(\mathrm{de1}\,x\,a\,y\left(x\right)\right)\:$

$\mathrm{de2}≔\mathrm{DEnormal}\left(\mathrm{de2}\,x\,y\left(x\right)\right)$

$\mathrm{de2}≔\left(50x^6-150x^5+150x^4-50x^3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(42x^4-84x^3+84x^2-42x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-42y\left(x\right)x^3+63y\left(x\right)x^2-105y\left(x\right)x+42y\left(x\right)=x\sin \left(x\right)$