\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch1}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull1}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty1}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull1}{\hidepaste}
\tab{5}\spadcommand{Dx: LODO(EXPR INT, f +-> D(f, x))\bound{Dxd }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty1}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty1}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch1}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty1}{\showpaste}
\tab{5}\spadcommand{Dx: LODO(EXPR INT, f +-> D(f, x))\bound{Dxd }}
\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch2}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull2}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty2}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull2}{\hidepaste}
\tab{5}\spadcommand{Dx := D()\free{Dxd }\bound{Dx }}
\indentrel{3}\begin{verbatim}
   (2)  D
Type: LinearOrdinaryDifferentialOperator(Expression(Integer),theMap(*1;anonymousFunction;0;initial;internal))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty2}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty2}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch2}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty2}{\showpaste}
\tab{5}\spadcommand{Dx := D()\free{Dxd }\bound{Dx }}
\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch3}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull3}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty3}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull3}{\hidepaste}
\tab{5}\spadcommand{Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1\free{Dx }\bound{Dop }}
\indentrel{3}\begin{verbatim}
                       3
         3    G     - x  + H
   (3)  D  + -- D + --------
              2         3
             x         x
Type: LinearOrdinaryDifferentialOperator(Expression(Integer),theMap(*1;anonymousFunction;0;initial;internal))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty3}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty3}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch3}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty3}{\showpaste}
\tab{5}\spadcommand{Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1\free{Dx }\bound{Dop }}
\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch4}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull4}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty4}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull4}{\hidepaste}
\tab{5}\spadcommand{firstCoefficients(n) ==
  phi := reduce(+, [subscript(s,[i])*exp(x)/x^i for i in 0..n])
  phi1 := retract((Dop(phi) * x^(n+3) / exp x))@Polynomial(Integer)
  phi2 := phi1 :: UnivariatePolynomial(x, Polynomial Integer)
  sys := [coefficient(phi2, (n+3-i)::NNI) for i in 2..n+1]
  solve(sys, [subscript(s,[i]) for i in 1..n])
}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty4}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty4}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch4}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty4}{\showpaste}
\tab{5}\spadcommand{firstCoefficients(n) ==
  phi := reduce(+, [subscript(s,[i])*exp(x)/x^i for i in 0..n])
  phi1 := retract((Dop(phi) * x^(n+3) / exp x))@Polynomial(Integer)
  phi2 := phi1 :: UnivariatePolynomial(x, Polynomial Integer)
  sys := [coefficient(phi2, (n+3-i)::NNI) for i in 2..n+1]
  solve(sys, [subscript(s,[i]) for i in 1..n])
}
\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch5}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull5}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty5}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull5}{\hidepaste}
\tab{5}\spadcommand{firstCoefficients(3)}
\indentrel{3}\begin{verbatim}
   (5)
   [
                                2
           s G       3 s H + s G  + 6 s G
            0           0     0        0
     [s  = ---, s  = --------------------,
       1    3    2            18
                                 3         2
           (9 s G + 54 s )H + s G  + 18 s G  + 72 s G
               0        0      0         0         0
      s  = ------------------------------------------]
       3                       162
     ]
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty5}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty5}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch5}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty5}{\showpaste}
\tab{5}\spadcommand{firstCoefficients(3)}
\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch6}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull6}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty6}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull6}{\hidepaste}
\tab{5}\spadcommand{firstCoefficients(4)}
\indentrel{3}\begin{verbatim}
   (6)
   [
                                2
           s G       3 s H + s G  + 6 s G
            0           0     0        0
     [s  = ---, s  = --------------------,
       1    3    2            18
                                 3         2
           (9 s G + 54 s )H + s G  + 18 s G  + 72 s G
               0        0      0         0         0
      s  = ------------------------------------------,
       3                       162

         s
          4
       = 
                   2          2
             27 s H  + (18 s G  + 378 s G + 1296 s )H
                 0          0          0          0
           + 
                4         3          2
             s G  + 36 s G  + 396 s G  + 1296 s G
              0         0          0           0
        /
           1944
       ]
     ]
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty6}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty6}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch6}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty6}{\showpaste}
\tab{5}\spadcommand{firstCoefficients(4)}
\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch7}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull7}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty7}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull7}{\hidepaste}
\tab{5}\spadcommand{firstCoefficients(7)}
\indentrel{3}\begin{verbatim}
   (7)
   [
                                2
           s G       3 s H + s G  + 6 s G
            0           0     0        0
     [s  = ---, s  = --------------------,
       1    3    2            18
                                 3         2
           (9 s G + 54 s )H + s G  + 18 s G  + 72 s G
               0        0      0         0         0
      s  = ------------------------------------------,
       3                       162

         s
          4
       = 
                   2          2
             27 s H  + (18 s G  + 378 s G + 1296 s )H
                 0          0          0          0
           + 
                4         3          2
             s G  + 36 s G  + 396 s G  + 1296 s G
              0         0          0           0
        /
           1944
       ,

         s
          5
       = 
                                 2
             (135 s G + 2268 s )H
                   0          0
           + 
                    3           2
             (30 s G  + 1350 s G  + 16416 s G + 38880 s )H
                  0           0            0           0
           + 
                5         4           3           2
             s G  + 60 s G  + 1188 s G  + 9504 s G
              0         0           0           0
           + 
             25920 s G
                    0
        /
           29160
       ,

         s
          6
       = 
                    3
             405 s H
                  0
           + 
                     2                          2
             (405 s G  + 18468 s G + 174960 s )H
                   0            0            0
           + 
                       4           3            2
                 45 s G  + 3510 s G  + 88776 s G
                     0           0            0
               + 
                 777600 s G + 1166400 s
                         0             0
            *
               H
           + 
                6         5           4            3
             s G  + 90 s G  + 2628 s G  + 27864 s G
              0         0           0            0
           + 
                      2
             90720 s G
                    0
        /
           524880
       ,

         s
          7
       = 
                                   3
             (2835 s G + 91854 s )H
                    0           0
           + 
                        3            2
                 945 s G  + 81648 s G  + 2082996 s G
                      0            0              0
               + 
                 14171760 s
                           0
            *
                2
               H
           + 
                       5           4             3
                 63 s G  + 7560 s G  + 317520 s G
                     0           0             0
               + 
                            2
                 5554008 s G  + 34058880 s G
                          0               0
            *
               H
           + 
                7          6           5            4
             s G  + 126 s G  + 4788 s G  + 25272 s G
              0          0           0            0
           + 
                        3               2
           - 1744416 s G  - 26827200 s G  - 97977600 s G
                      0               0               0
        /
           11022480
       ]
     ]
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty7}
\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty7}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch7}
\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty7}{\showpaste}
\tab{5}\spadcommand{firstCoefficients(7)}
\end{paste}\end{patch}

