> 
> # Tehtava 5
> 
> # a-kohta
> # Kaytetaan symboleina kirjaimia v, w ja K 
> dya:=diff(y(t),t$2)+v^2*y(t)=K*sin(w*t);

                          /   2      \
                          |  d       |    2
                   dya := |----- y(t)| + v  y(t) = K sin(w t)
                          |   2      |
                          \ dt       /
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> readlib(laplace):
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> ldya:=laplace(dya,t,s);

ldya :=

                                                 2                         K w
   (laplace(y(t), t, s) s - y(0)) s - D(y)(0) + v  laplace(y(t), t, s) = -------
                                                                          2    2
                                                                         s  + w
--------------------------------------------------------------------------------
> ldya:=subs({y(0)=0,D(y)(0)=0},ldya);

                                     2    2                         K w
        ldya := laplace(y(t), t, s) s  + v  laplace(y(t), t, s) = -------
                                                                   2    2
                                                                  s  + w
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> lratka:=solve(ldya,laplace(y(t),t,s));

                                            K w
                          lratka := -------------------
                                      2    2    2    2
                                    (s  + w ) (s  + v )
--------------------------------------------------------------------------------
> ratka:=invlaplace(lratka,s,t);

                                 K sin(w t)    K w sin(v t)
                      ratka := - ---------- + -------------
                                     2    2       2    2
                                  - v  + w    (- v  + w ) v
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> 
> # b-kohta
> y:='y': v:='v': w:='w': K:='K':
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> dyb:=diff(y(t),t$2)+v^2*y(t)=K*sin(v*t);

                          /   2      \
                          |  d       |    2
                   dyb := |----- y(t)| + v  y(t) = K sin(v t)
                          |   2      |
                          \ dt       /
--------------------------------------------------------------------------------
> ldyb:=laplace(dyb,t,s);

ldyb :=

                                                 2                         K v
   (laplace(y(t), t, s) s - y(0)) s - D(y)(0) + v  laplace(y(t), t, s) = -------
                                                                          2    2
                                                                         s  + v
--------------------------------------------------------------------------------
> ldyb:=subs({y(0)=0,D(y)(0)=0},ldyb);

                                     2    2                         K v
        ldyb := laplace(y(t), t, s) s  + v  laplace(y(t), t, s) = -------
                                                                   2    2
                                                                  s  + v
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> lratkb:=solve(ldyb,laplace(y(t),t,s));

                                            K v
                              lratkb := ----------
                                          2    2 2
                                        (s  + v )
--------------------------------------------------------------------------------
> ratkb:=invlaplace(lratkb,s,t);

                                 t cos(v t) K       K sin(v t)
                  ratkb := - 1/2 ------------ + 1/2 ----------
                                       v                 2
                                                        v
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>