> A:=dsolve({diff(y(t),t,t)+w0^2*y(t)=cos(w*t),y(0)=0,D(y)(0)=0},y(t));

                              cos(w0 t)   cos(w t)
                  A := y(t) = --------- - ---------
                                 2    2      2    2
                              -w0  + w    -w0  + w

> w0:=10;w:=1;plot(rhs(A),t=0..2*Pi);

                               w0 := 10


                                w := 1

> rhs(A);w:='w';w;w0:='w0';w0;
> 

  -3/5 exp(-t) sin(t) - 1/5 exp(-t) cos(t) + 2/5 sin(t) + 1/5 cos(t)


                                w := w


                                  w


                               w0 := w0


                                  w0

> expand(cos(m+n)-cos(m-n));

                           -2 sin(m) sin(n)

> plot(sin(20*t)*sin(t),t=0..2*Pi);
> A:=dsolve({diff(y(t),t,t)+2*diff(y(t),t)+2*y(t)=cos(w*t),y(0)=0,D(y)(0)=0},y(t));

                                 2                              2
                exp(-t) sin(t) (w  + 2)   exp(-t) cos(t) (-2 + w )
  A := y(t) = - ----------------------- + ------------------------
                             4                          4
                        4 + w                      4 + w

                                        2
           2 cos(w t) + 2 sin(w t) w - w  cos(w t)
         + ---------------------------------------
                                4
                           4 + w

> plot(sqrt((2*w)^2+(2-w^2)^2)/(4+w^4),w=0..4);
> A:=dsolve({diff(y(t),t,t)+0.2*diff(y(t),t)+2*y(t)=cos(w*t),y(0)=0,D(y)(0)=0},y(t));

                                      1/2
                          t        199    t     1/2   2
                   exp(- ----) sin(--------) 199    (w  + 2)
               25         10          10
  A := y(t) = ---- -----------------------------------------
               199                      2       4
                              100 - 99 w  + 25 w

                                 1/2
                     t        199    t         2
           25 exp(- ----) cos(--------) (-2 + w )
                     10          10
         + --------------------------------------
                              2       4
                    100 - 99 w  + 25 w

                              2
           5 sin(w t) w - 25 w  cos(w t) + 50 cos(w t)
         + -------------------------------------------
                                 2       4
                       100 - 99 w  + 25 w

> plot(sqrt((5*w)^2+(50-25*w^2)^2)/(100-99*w^2+25*w^4),w=0..4);
> A:=dsolve({diff(y(t),t,t)+0.02*diff(y(t),t)+2*y(t)=cos(w*t),y(0)=0,D(y)(0)=0},y(t));

                                         1/2
                            t       19999    t       1/2   2
                     exp(- ---) sin(----------) 19999    (w  + 2)
               2500        100         100
  A := y(t) = ------ --------------------------------------------
               19999                        2         4
                              10000 - 9999 w  + 2500 w

                                    1/2
                       t       19999    t         2
           2500 exp(- ---) cos(----------) (-2 + w )
                      100         100
         + -----------------------------------------
                                 2         4
                   10000 - 9999 w  + 2500 w

                                                 2
           5000 cos(w t) + 50 sin(w t) w - 2500 w  cos(w t)
         + ------------------------------------------------
                                    2         4
                      10000 - 9999 w  + 2500 w

> plot(sqrt((50*w)^2+(5000-2500*w^2)^2)/(10000-9999*w^2+2500*w^4),w=0..4);
>