O15-C912
> restart:
> f:=x->abs(sin(x))^5;

                                            5
                        f := x -> | sin(x) |

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

                                Pi
                               /
                              |
                              |    f(x) cos(2 n x) dx
                              |
                             /
                               0
                 a := n -> 2 ------------------------
                                        Pi


                2
  -480 cos(Pi n) /((2 n + 5) (2 n - 5) (2 n + 3) (2 n - 3) (2 n + 1)

        (2 n - 1) Pi)


> sompart:=proc(N,t)
> if N=0 then a(0)/2 else sompart(N-1,t)+a(N)*cos(2*N*t) fi;
> end;

  sompart := proc(N, t)
    if N = 0 then 1/2*a(0)
    else sompart(N - 1, t) + a(N)*cos(2*N*t)
    fi
end

> s:=seq(sompart(N,t),N=0..3);

       16  1    16  1     32 cos(2 t)
  s := -- ----, -- ---- - -- --------,
       15  Pi   15  Pi    21    Pi

        16  1     32 cos(2 t)   32 cos(4 t)
        -- ---- - -- -------- + -- --------,
        15  Pi    21    Pi      63    Pi

        16  1     32 cos(2 t)   32 cos(4 t)   32  cos(6 t)
        -- ---- - -- -------- + -- -------- - --- --------
        15  Pi    21    Pi      63    Pi      693    Pi

> plot([s,f(t)],t=0..2*Pi);

> s:=dsolve(diff(x(t),t,t)+x(t)-cos(n*t),x(t));

              /    sin((-1 + n) t)       sin((1 + n) t)\
  s := x(t) = |1/2 --------------- + 1/2 --------------| sin(t)
              \        -1 + n                1 + n     /

           /    cos((1 + n) t)       cos((-1 + n) t)\
         + |1/2 -------------- - 1/2 ---------------| cos(t)
           \        1 + n                -1 + n     /

         + _C1 sin(t) + _C2 cos(t)

> dsolve(diff(x(t),t,t)+x(t)-cos(t),x(t));

                                                        2
  x(t) = (1/2 sin(t) cos(t) + 1/2 t) sin(t) - 1/2 sin(t)  cos(t)

         + _C1 sin(t) + _C2 cos(t)

> g:=a(0)/2-sum(a(n)*cos(2*n*t)/(-1+4*n^2),n=1..100):gg:=unapply(g,t):evalf(gg(0));(D(gg))(0);

                             .4908738521


                                  0

> s1:=dsolve({diff(x(t),t,t)+x(t)-abs(sin(t)^5),x(0)=.4908738521, D(x)(0)=0},x(t));

                         2           5                  5
  s1 := x(t) = 1/6 sin(t)  | sin(t) |  - 1/48 | sin(t) |  (

                 5                   3
        -8 sin(t)  cos(t) - 10 sin(t)  cos(t) - 15 sin(t) cos(t)

                          /       5
         + 15 t) cos(t)  /  sin(t)  + .4908738521 cos(t)
                        /

> g1:=subs(s1,x(t));

                  2           5                  5           5
  g1 := 1/6 sin(t)  | sin(t) |  - 1/48 | sin(t) |  (-8 sin(t)  cos(t)

                    3                                            /
         - 10 sin(t)  cos(t) - 15 sin(t) cos(t) + 15 t) cos(t)  /
                                                               /

              5
        sin(t)  + .4908738521 cos(t)

> plot([g,g1],t=0..Pi);

>