O15-C910
> restart:
> ed:=-x^2*diff(y(x),x,x)-2*x*diff(y(x),x)+y(x);

                       / 2      \
                     2 |d       |       /d      \
             ed := -x  |--- y(x)| - 2 x |-- y(x)| + y(x)
                       |  2     |       \dx     /
                       \dx      /

> dsolve(ed,y(x));

                        / 1/2      \        /   1/2      \
                        |5         |        |  5         |
                        |---- - 1/2|        |- ---- - 1/2|
                        \ 2        /        \   2        /
            y(x) = _C1 x             + _C2 x

> ed1:=-x^2*diff(y(x),x,x)-2*x*diff(y(x),x)+y(x)=arctan(x);

                  / 2      \
                2 |d       |       /d      \
       ed1 := -x  |--- y(x)| - 2 x |-- y(x)| + y(x) = arctan(x)
                  |  2     |       \dx     /
                  \dx      /

> sol1:=dsolve(ed1,y(x));

                  / 1/2      \        /   1/2      \
                  |5         |        |  5         |
                  |---- - 1/2|        |- ---- - 1/2|            /
                  \ 2        /        \   2        /            |
  sol1 := y(x) = x             _C2 + x               _C1 + 1/20 |
                                                                \

                                                      1/2
               1/2                       2           5
        -(5 + 5   ) x csgn(x) LerchPhi(-x , 1, 3/4 - ----)
                                                      4

                                                   1/2
                  1/2                       2     5
         + (-5 + 5   ) x csgn(x) LerchPhi(-x , 1, ---- + 3/4)
                                                   4

                               \
                               |
         + 20 arctan(csgn(x) x)| csgn(x)
                               /

> f:=subs(sol1,y(x));

        / 1/2      \        /   1/2      \
        |5         |        |  5         |
        |---- - 1/2|        |- ---- - 1/2|            /
        \ 2        /        \   2        /            |
  f := x             _C2 + x               _C1 + 1/20 |
                                                      \

                                                      1/2
               1/2                       2           5
        -(5 + 5   ) x csgn(x) LerchPhi(-x , 1, 3/4 - ----)
                                                      4

                                                   1/2
                  1/2                       2     5
         + (-5 + 5   ) x csgn(x) LerchPhi(-x , 1, ---- + 3/4)
                                                   4

                               \
                               |
         + 20 arctan(csgn(x) x)| csgn(x)
                               /

> g:=subs({_C1=0,_C2=0},f);# Solution particulière sous forme intégrale

            /                                              1/2
            |       1/2                       2           5
  g := 1/20 |-(5 + 5   ) x csgn(x) LerchPhi(-x , 1, 3/4 - ----)
            \                                              4

                                                   1/2
                  1/2                       2     5
         + (-5 + 5   ) x csgn(x) LerchPhi(-x , 1, ---- + 3/4)
                                                   4

                               \
                               |
         + 20 arctan(csgn(x) x)| csgn(x)
                               /

> p:=convert(series(g,x=0,11),polynom);# Partie régulière du DL10 en 0

            /        1/2                   1/2                      \
            |  (5 + 5   ) csgn(x)   (-5 + 5   ) csgn(x)             |
  p := 1/20 |- ------------------ + ------------------- + 20 csgn(x)|
            |             1/2            1/2                        |
            |            5              5                           |
            |      3/4 - ----           ---- + 3/4                  |
            \             4              4                          /

        csgn(x) x + 1/20

        /      1/2                   1/2                         \
        |(5 + 5   ) csgn(x)   (-5 + 5   ) csgn(x)               3|
        |------------------ - ------------------- - 20/3 csgn(x) |
        |           1/2                  1/2                     |
        |          5                    5                        |
        |    7/4 - ----           7/4 + ----                     |
        \           4                    4                       /

                 3
        csgn(x) x  + 1/20

        /        1/2                   1/2                      \
        |  (5 + 5   ) csgn(x)   (-5 + 5   ) csgn(x)            5|
        |- ------------------ + ------------------- + 4 csgn(x) |
        |             1/2            1/2                        |
        |            5              5                           |
        |     11/4 - ----           ---- + 11/4                 |
        \             4              4                          /

                 5
        csgn(x) x  + 1/20

        /      1/2                   1/2                         \
        |(5 + 5   ) csgn(x)   (-5 + 5   ) csgn(x)               7|
        |------------------ - ------------------- - 20/7 csgn(x) |
        |           1/2            1/2                           |
        |          5              5                              |
        |   15/4 - ----           ---- + 15/4                    |
        \           4              4                             /

                 7
        csgn(x) x  + 1/20

        /        1/2                   1/2                         \
        |  (5 + 5   ) csgn(x)   (-5 + 5   ) csgn(x)               9|
        |- ------------------ + ------------------- + 20/9 csgn(x) |
        |             1/2            1/2                           |
        |            5              5                              |
        |     19/4 - ----           ---- + 19/4                    |
        \             4              4                             /

                 9
        csgn(x) x

> map(simplify,p);seq(cos((p+1)*Pi)/(2*p+1)/(4*p^2+6*p+1),p=0..4);# On retrouve les premiers coeffs de la solution DSE a_(2p+1)=(-1)^(p+1)/(2p+1)/(4p^2+6p+1) (R=1 et convergence en R et -R)

                       3          5          7          9
            -x + 1/33 x  - 1/145 x  + 1/385 x  - 1/801 x


                                -1          -1
                      -1, 1/33, ---, 1/385, ---
                                145         801

>