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

Take the differential ring $C\left(x\right)$ $\left[\mathrm{Dx}\right]$:
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$\mathrm{\_Envdiffopdomain}\u2254\left[\mathrm{Dx}\,x\right]$

${\mathrm{\_Envdiffopdomain}}{\u2254}\left[{\mathrm{Dx}}{\,}{x}\right]$
 (1) 
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$\mathrm{L1}\u2254\left({x}^{2}x\right){\mathrm{Dx}}^{2}+\left(ax+bx+xc\right)\mathrm{Dx}+ab$

${\mathrm{L1}}{\u2254}\left({{x}}^{{2}}{}{x}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({a}{}{x}{+}{b}{}{x}{}{c}{+}{x}\right){}{\mathrm{Dx}}{+}{a}{}{b}$
 (2) 
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$\mathrm{L2}\u2254\mathrm{subs}\left(c=c+1\,\mathrm{L1}\right)$

${\mathrm{L2}}{\u2254}\left({{x}}^{{2}}{}{x}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({a}{}{x}{+}{b}{}{x}{}{c}{+}{x}{}{1}\right){}{\mathrm{Dx}}{+}{a}{}{b}$
 (3) 
Compute a basis for the homomorphisms$r:V\left(\mathrm{L1}\right)\to V\left(\mathrm{L2}\right)$.
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$v\u2254\mathrm{Homomorphisms}\left(\mathrm{L1}\,\mathrm{L2}\right)$

${v}{\u2254}\left[\left({x}{}{1}\right){}{\mathrm{Dx}}{}{c}{+}{a}{+}{b}\right]$
 (4) 
Since this basis has precisely one element, there is, up to multiplication by constants, precisely one map $V\left(\mathrm{L1}\right)\to V\left(\mathrm{L2}\right)$ that can be presented by an operator $r\in C\left(x\right)$ $\left[\mathrm{Dx}\right]$.
In the following example, every linear map $V\left(\mathrm{L1}\right)\to V\left(\mathrm{L2}\right)$ can be presented by an operator. Thus, the dimension of all such maps will be $\mathrm{order}\left(\mathrm{L1}\right)\mathrm{order}\left(\mathrm{L2}\right)=32=6$. Since the output is a basis of these maps, it must have 6 elements.
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$\mathrm{L1}\u2254{\mathrm{Dx}}^{3}\;$$\mathrm{L2}\u2254{\mathrm{Dx}}^{2}$

${\mathrm{L1}}{\u2254}{{\mathrm{Dx}}}^{{3}}$
 
${\mathrm{L2}}{\u2254}{{\mathrm{Dx}}}^{{2}}$
 (5) 
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$\mathrm{Homomorphisms}\left(\mathrm{L1}\,\mathrm{L2}\right)$

$\left[{\mathrm{Dx}}{\,}{x}{}{\mathrm{Dx}}{}{2}{\,}{{\mathrm{Dx}}}^{{2}}{\,}{x}{}{{\mathrm{Dx}}}^{{2}}{}{\mathrm{Dx}}{\,}{{x}}^{{2}}{}{{\mathrm{Dx}}}^{{2}}{}{2}{}{x}{}{\mathrm{Dx}}{+}{2}{\,}{{x}}^{{3}}{}{{\mathrm{Dx}}}^{{2}}{}{2}{}{{x}}^{{2}}{}{\mathrm{Dx}}{+}{2}{}{x}\right]$
 (6) 