Question:
Differential root reduce uses the font color of the stylesheet and a default white background. How to change the color of the background ?
Why :
I am using a dark style sheet where text is written in white on a black background. However, the white font and white background of the output makes it illegible like in the image below:
Both DifferentialRoot and DifferenceRoot use the same template for their box structure, which is named "HolonomicDisplay".
Therefore, the styling is set by the stylesheet, which you can access with CurrentValue[{StyleDefinitions, "HolonomicDisplay"}], and also change in the usual manner of changing stylesheet. You can also set it programatically:
displayFunction = (TemplateBoxOptionsDisplayFunction /. CurrentValue[{StyleDefinitions, "HolonomicDisplay"}]) /. (Background -> _) -> (Background -> Yellow); SetOptions[SelectedNotebook[], StyleDefinitions -> Notebook[{Cell[StyleData[StyleDefinitions -> "Default.nb"]], Cell[StyleData["HolonomicDisplay"], TemplateBoxOptions -> {InterpretationFunction -> (#2 &), DisplayFunction -> displayFunction}]}]]; 
It seems that TemplateBoxOptionsDisplayFunction was changed to TemplateBoxOptions in 13.1. Therefore, you should instead use:
displayFunction = (TemplateBoxOptions /. CurrentValue[{StyleDefinitions, "HolonomicDisplay"}]) /. (Background -> _) -> (Background -> Yellow); 
ToBoxes when you want to fiddle with Mathematica designs. $\endgroup$ DifferenceRoot from the documentation! Didn't even notice it was the same as yours :-) $\endgroup$