> 
> # Tehtava 3
> readlib(laplace):
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> alias(H=Heaviside);

                                      I, H
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> dy:=diff(x(t),t$2)+2*diff(x(t),t)+x(t)=Dirac(t)-H(t-2*Pi);

             /   2      \
             |  d       |     /  d      \
       dy := |----- x(t)| + 2 |---- x(t)| + x(t) = Dirac(t) - H(t - 2 Pi)
             |   2      |     \ dt      /
             \ dt       /
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> ldy:=laplace(dy,t,s);

   ldy := (laplace(x(t), t, s) s - x(0)) s - D(x)(0) + 2 laplace(x(t), t, s) s

                                             exp(- 2 Pi s)
        - 2 x(0) + laplace(x(t), t, s) = 1 - -------------
                                                   s
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> ldy:=subs({x(0)=0,D(x)(0)=0},ldy);

                            2
ldy := laplace(x(t), t, s) s  + 2 laplace(x(t), t, s) s + laplace(x(t), t, s) =

        exp(- 2 Pi s)
    1 - -------------
              s
--------------------------------------------------------------------------------
> lratk:=solve(ldy,laplace(x(t),t,s));

                                          exp(- 2 Pi s)
                                    - 1 + -------------
                                                s
                         lratk := - -------------------
                                         2
                                        s  + 2 s + 1
--------------------------------------------------------------------------------
> ratk:=invlaplace(lratk,s,t);

               t
               / / -----            \
              |  |  \     exp(_r _U)|
   ratk := -  |  |   )    ----------| (- Dirac(t - _U) + H(t - _U - 2 Pi)) d_U
              |  |  /      2 _r + 2 |
             /   | -----            |
             0   \_r = %1           /

                                     2
%1 :=                       RootOf(_Z  + 2 _Z + 1)
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> # HUI   HIRVITYS  !!!!!!!!!!!!!!!!
> # Yritetaan uudestaan
> lratk2:=1/(s+1)^2-exp(-2*Pi*s)/(s*(s+1)^2);

                                     1      exp(- 2 Pi s)
                       lratk2 := -------- - -------------
                                        2              2
                                 (s + 1)      s (s + 1)
--------------------------------------------------------------------------------
> ratk2:=invlaplace(lratk2,s,t);

 ratk2 :=

     t exp(- t) - H(t - 2 Pi) (1 - (t - 2 Pi) exp(- t + 2 Pi) - exp(- t + 2 Pi))
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> # No, nyt nayttaa paremmalta!
> # Tekijoihinjaon olisi ehka voinut tehda Maplellakin, mutta en loytanyt keinoa tahan
> with(plots): plot(ratk2,t=0..25);
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>