> 
> # Tehtava 4
> 
> dy:=diff(y(t),t$2)+3*diff(y(t),t)+2*y(t);

                         /   2      \
                         |  d       |     /  d      \
                   dy := |----- y(t)| + 3 |---- y(t)| + 2 y(t)
                         |   2      |     \ dt      /
                         \ dt       /
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> alias(H=Heaviside);

                                      I, H
> # a-kohta
> dya:=dy=H(t)-H(t-1);

                /   2      \
                |  d       |     /  d      \
         dya := |----- y(t)| + 3 |---- y(t)| + 2 y(t) = H(t) - H(t - 1)
                |   2      |     \ dt      /
                \ dt       /
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> dsolve({dya,y(0)=0,D(y)(0)=0},y(t),method=laplace);

            y(t) = 1/2 + 1/2 exp(- 2 t) - exp(- t)

                 - H(t - 1) (1/2 + 1/2 exp(- 2 t + 2) - exp(- t + 1))
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> # Kuten viime viikolla, ratkaistaan y(t) uudestaan solve:lla
> y(t):=solve(",y(t));

           y(t) := 1/2 + 1/2 exp(- 2 t) - exp(- t) - 1/2 H(t - 1)

                - 1/2 H(t - 1) exp(- 2 t + 2) + H(t - 1) exp(- t + 1)
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> plot({y(t),H(t)-H(t-1)},t=0..1);
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> 
> # b-kohta
> y:='y': dyb:=dy=t*(H(t)-H(t-1));

              /   2      \
              |  d       |     /  d      \
       dyb := |----- y(t)| + 3 |---- y(t)| + 2 y(t) = t (H(t) - H(t - 1))
              |   2      |     \ dt      /
              \ dt       /
--------------------------------------------------------------------------------
> dsolve({dyb,y(0)=0,D(y)(0)=0},y(t),method=laplace);

        y(t) = 1/2 t - 3/4 - 1/4 exp(- 2 t) + exp(- t)

             - H(t - 1) (1/2 + 1/2 exp(- 2 t + 2) - exp(- t + 1))

             - H(t - 1) (1/2 t - 5/4 - 1/4 exp(- 2 t + 2) + exp(- t + 1))
--------------------------------------------------------------------------------
> y(t):=solve(",y(t));

         y(t) := 1/2 t - 3/4 - 1/4 exp(- 2 t) + exp(- t) + 3/4 H(t - 1)

              - 1/4 H(t - 1) exp(- 2 t + 2) - 1/2 H(t - 1) t
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> plot({y(t),t*(H(t)-H(t-1))},t=0..2);
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> 
> # c-kohta
> y:='y': dyc:=dy=10*(H(t)-H(t-5)+H(t-10)-H(t-15)+H(t-20)-H(t-25)); 

             /   2      \
             |  d       |     /  d      \
      dyc := |----- y(t)| + 3 |---- y(t)| + 2 y(t) = 10 H(t) - 10 H(t - 5)
             |   2      |     \ dt      /
             \ dt       /

           + 10 H(t - 10) - 10 H(t - 15) + 10 H(t - 20) - 10 H(t - 25)
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> dsolve({dyc,y(0)=0,D(y)(0)=0},y(t),method=laplace);

         y(t) = 5 + 5 exp(- 2 t) - 10 exp(- t)

              - 10 H(t - 5) (1/2 + 1/2 exp(- 2 t + 10) - exp(- t + 5))

              + 10 H(t - 10) (1/2 + 1/2 exp(- 2 t + 20) - exp(- t + 10))

              - 10 H(t - 15) (1/2 + 1/2 exp(- 2 t + 30) - exp(- t + 15))

              + 10 H(t - 20) (1/2 + 1/2 exp(- 2 t + 40) - exp(- t + 20))

              - 10 H(t - 25) (1/2 + 1/2 exp(- 2 t + 50) - exp(- t + 25))
--------------------------------------------------------------------------------
> y(t):=solve(",y(t));

y(t) := 5 + 5 exp(- 2 t) - 10 exp(- t) - 5 H(t - 5) - 5 H(t - 5) exp(- 2 t + 10)

     + 10 H(t - 5) exp(- t + 5) + 5 H(t - 10) + 5 H(t - 10) exp(- 2 t + 20)

     - 10 H(t - 10) exp(- t + 10) - 5 H(t - 15) - 5 H(t - 15) exp(- 2 t + 30)

     + 10 H(t - 15) exp(- t + 15) + 5 H(t - 20) + 5 H(t - 20) exp(- 2 t + 40)

     - 10 H(t - 20) exp(- t + 20) - 5 H(t - 25) - 5 H(t - 25) exp(- 2 t + 50)

     + 10 H(t - 25) exp(- t + 25)
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> plot({y(t),10*(H(t)-H(t-5)+H(t-10)-H(t-15)+H(t-20)-H(t-25))},t=0..25);
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>