TeXForm for bold symbols and cells

You can use a replacement rule for the example you posted. For each of these rules, the lhs is a Mathematica expression and the rhs is the styled string you want to see in the TeX, representing that expression.

TeXForm[HoldForm[y''[t] + G[y'[t], y[t]] == F[t]]] 

(* y''(t)+G\left(y'(t),y(t)\right)=F(t) *)

This is basically just a string. Now just replace G,y,F with Style[_,Bold]:

%2 /. {G -> Style["G", Bold], y -> Style["y", Bold], F -> Style["F", Bold]} 

(It doesn't render well in the browser so I'll post a screenshot.)

If you also want to use bold type for the "primes" on the derivatives, you need to write a pattern which matches the derivative operator in the code. In this case, the nth derivative is Derivative[n][y][t]. Since you're mapping to a string, each 'n' must be written separately. To get the arguments in normal weight, use a second "Text" style box for each "[t]".

%2 /. {G -> Style["G", Bold], y -> Style["y", Bold], F -> Style["F", Bold], Derivative[1][y][t] :> Style["y'", Bold] Style["[t]", "Text"], Derivative[2][y][t] :> Style["y''", Bold] Style["[t]", "Text"]} 

See TeXForm help. No Mathematica format boxes are translated, only single character symbols as italics and word symbols as \text{} boxes are used in translation.

This yields a possibility to modify an expressions TeXForm, as a string, e.g. by

 raw = "$$" <> ToString[TeXForm[y''[x] == Sin[x]]] <> "$$" bold = StringReplace[raw, "y" -> "\\mathbf{y}"] 

$$\mathbf{y}''(x)=\sin (x)$$

 Import[ StringToStream[bold], "LaTeX" ] Notebook[{Cell[TextData[{}], "Text"], Cell[BoxData[FormBox[RowBox[{SuperscriptBox[StyleBox["y", FontSlant -> "Bold"],"\[Prime]\[Prime]"], RowBox[{"(", StyleBox["x", "TI"], ")"}],"\[LongEqual]", "sin", RowBox[{"(", StyleBox["x", "TI"], ")"}]}], TraditionalForm]], "NumberedEquation"]}, StyleDefinitions -> "Default.nb"]