If the input L is the differential operator $a_0{\mathrm{Dx}}^0+...+a_n{\mathrm{Dx}}^n$, then the output is ${\left(-1\right)}^n\left(\mathrm{mult}\left({\left(-\mathrm{Dx}\right)}^0,a_0\right)+...+\mathrm{mult}\left({\left(-\mathrm{Dx}\right)}^n,a_n\right)\right)$.

The adjoint has the following properties: adjoint(adjoint(L)) = L and mult(adjoint(L), adjoint(M)) = adjoint(mult(M,L)). So applying the adjoint results in switching the order of multiplication; the adjoint is an anti-automorphism of C(x)[Dx], multiplied by ${\left(\mathrm{-1}\right)}^n$.

The argument domain describes the differential algebra. If this argument is the list $\left[\mathrm{Dx}\,x\right]$, then the differential operators are notated with the symbols $\mathrm{Dx}$ and $x$. They are viewed as elements of the differential algebra C(x)[Dx] where $C$ is the field of constants.

If the argument domain is omitted, then the differential specified by the environment variable _Envdiffopdomain is used. If this environment variable is not set then the argument domain may not be omitted.

Instead of a differential operator, the input can also be a linear homogeneous differential equation eqn. In this case the second argument dvar must be the dependent variable.

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

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

$\mathrm{\_Envdiffopdomain}≔\left[\mathrm{Dx}\,x\right]$

$\mathrm{\_Envdiffopdomain}≔\left[\mathrm{Dx}\,x\right]$

$L≔{\mathrm{Dx}}^5+2x^3{\mathrm{Dx}}^3+6x^2{\mathrm{Dx}}^2+6x\mathrm{Dx}+x^6\mathrm{Dx}+1$

$L≔x^6\mathrm{Dx}+2x^3{\mathrm{Dx}}^3+{\mathrm{Dx}}^5+6x^2{\mathrm{Dx}}^2+6x\mathrm{Dx}+1$

$\mathrm{adjoint}\left(L\right)$

${\mathrm{Dx}}^5+2x^3{\mathrm{Dx}}^3+12x^2{\mathrm{Dx}}^2+x^6\mathrm{Dx}+18x\mathrm{Dx}+6x^5+5$

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

${\mathrm{Dx}}^5+2x^3{\mathrm{Dx}}^3+6x^2{\mathrm{Dx}}^2+x^6\mathrm{Dx}+6x\mathrm{Dx}+1$

$\mathrm{adjoint}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)+y\left(x\right)\,y\left(x\right)\right)$

$-y\left(x\right)+\frac{\ⅆ}{\ⅆx}y\left(x\right)$