O17-963
> restart;
> edo:=9*(1-x^2)*diff(y(x),x$2)-9*x*diff(y(x),x)+y(x);

                           / 2      \
                        2  |d       |       /d      \
         edo := 9 (1 - x ) |--- y(x)| - 9 x |-- y(x)| + y(x)
                           |  2     |       \dx     /
                           \dx      /

> sol:=dsolve({edo,y(0)=1/2},y(x));

    sol := y(x) = _C1 sin(1/3 arcsin(x)) + 1/2 cos(1/3 arcsin(x))

> g:=subs(sol,y(x));

         g := _C1 sin(1/3 arcsin(x)) + 1/2 cos(1/3 arcsin(x))

> limit(g,x=1);limit(g,x=-1);

                                     1/2
                              _C1   3
                              --- + ----
                               2     4


                                      1/2
                               _C1   3
                             - --- + ----
                                2     4

> gprim:=diff(g,x);

                  _C1 cos(1/3 arcsin(x))       sin(1/3 arcsin(x))
     gprim := 1/3 ---------------------- - 1/6 ------------------
                             2 1/2                      2 1/2
                       (1 - x )                   (1 - x )

> l:=limit(gprim*sqrt(1-x),x=1);

                            1/2      1/2    1/2
                           2    _C1 3      2
                      l := ------------- - ----
                                12          24

> series(gprim,x=1,2);

                1/2  1/2           1/2    1/2
   -1/12 I _C1 3    2    + 1/24 I 2      3      _C1            1/2
   ----------------------------------- + ---- + --- + O((x - 1)   )
                      1/2                 36    18
               (x - 1)

> cste:=solve(l,_C1);

                                      1/2
                                     3
                             cste := ----
                                      6

> vraig:=subs(_C1=cste,g);

                  1/2
    vraig := 1/6 3    sin(1/3 arcsin(x)) + 1/2 cos(1/3 arcsin(x))

> vraigprim:=diff(vraig,x);

                     1/2
                    3    cos(1/3 arcsin(x))       sin(1/3 arcsin(x))
  vraigprim := 1/18 ----------------------- - 1/6 ------------------
                                2 1/2                      2 1/2
                          (1 - x )                   (1 - x )

> series(vraigprim,x=-1,2);

                      1/2         1/2
                     2           3               1/2
                 ------------- - ---- + O((x + 1)   )
                           1/2    54
                 12 (x + 1)

> vraigsec:=diff(vraig,x$2);

                     1/2
                    3    sin(1/3 arcsin(x))
  vraigsec := -1/54 -----------------------
                                 2
                            1 - x

                 1/2
                3    cos(1/3 arcsin(x)) x        cos(1/3 arcsin(x))
         + 1/18 ------------------------- - 1/18 ------------------
                             2 3/2                          2
                       (1 - x )                        1 - x

               sin(1/3 arcsin(x)) x
         - 1/6 --------------------
                         2 3/2
                   (1 - x )

> series(vraigsec,x=1,4);

        1/2       1/2                1/2        2
     8 3      56 3    (x - 1)   320 3    (x - 1)             (5/2)
   - ------ + --------------- - ----------------- + O((x - 1)     )
      729          6561               59049

>