> restart;
O14-055
> u:=x->Pi^2/(12)-x^2/4;

> a:=n->int(u(x)*cos(n*x),x=0..Pi)*2/Pi;assume(n,integer);a(n);


                                     n~
                                 (-1)
                               - ------
                                    2
                                  n~

Par DES de 1/n^2/(n^2+1), il vient g=f+g"

> eq:=y(x)=diff(y(x),x,x)+u(x);

                              / 2      \     2     2
                              |d       |   Pi     x
                 eq := y(x) = |--- y(x)| + --- - ----
                              |  2     |   12     4
                              \dx      /

> sol:=dsolve(eq,y(x));

                                                       2     2
                                                     Pi     x
      sol := y(x) = exp(x) _C2 + exp(-x) _C1 - 1/2 + --- - ----
                                                     12     4

> g1:=subs(sol,y(x));

                                                   2     2
                                                 Pi     x
          g1 := exp(x) _C2 + exp(-x) _C1 - 1/2 + --- - ----
                                                 12     4

> sys:={subs(x=0,diff(g1,x)),subs(x=Pi,diff(g1,x))};

                                       Pi
 sys := {exp(Pi) _C2 - exp(-Pi) _C1 - ----, exp(0) _C2 - exp(0) _C1}
                                       2

> g2:=subs(solve(sys),g1);

                                                                 2
            exp(x) Pi exp(Pi)       exp(-x) Pi exp(Pi)         Pi
  g2 := 1/2 ----------------- + 1/2 ------------------ - 1/2 + ---
                     2                        2                12
              exp(Pi)  - 1             exp(Pi)  - 1

             2
            x
         - ----
            4

> a2:=n->int(g2*cos(n*x),x=0..Pi)*2/Pi;assume(n,integer);a2(n);


                                    n~
                                (-1)
                           - -------------
                                    2    2
                             (1 + n~ ) n~

> evalf(subs(x=Pi,diff(g2,x,x)));

                             1.076674047

> evalf(sum(1/(n^2+1),n=1..infinity));

> evalf(subs(x=Pi,g2)-subs(x=Pi,diff(g2,x,x))+Pi^2/6);

                                  0.