Lösungen

1

Abgabe

2

Remove["Global`*"]

a

(LaplaceTransform[y''[t] + a y'[t] + b y[t], t, 
            s]/.{y[0] -> y0,y'[0]->yS0,LaplaceTransform[y[t], t, s] -> Y[s]})==(LaplaceTransform[
            f[t]] /. {LaplaceTransform[f[t]] -> F[s]})

b

Solve[-s y0 - yS0 + b Y[s] + (s^2) Y[s] + 
          a (-y0 + s Y[s]) == F[s], { Y[s]}]

((((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                        y[0] == y0, y'[0] == yS0}, y, t] // 
                    Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> k}) //.  
            (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 1/(2 k)) // 
        Simplify)

c

InverseLaplaceTransform[
      Evaluate[(((a y0 + s y0 + yS0 + F[s])/(b + a s + (s^2))) /. {a -> 1, b -> 1, F[s] -> 0, y0 -> 1, yS0 -> 0})], s, 
      t]

((DSolve[{y''[t] + 1  y'[t] + 1 y[t] == 0, 
                y[0] == 1, y'[0] == 0}, y, t] // 
            Flatten)) // Simplify

((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                  y[0] == y0, y'[0] == yS0}, y, t] // 
              Flatten) /. {a -> 1, b -> 1, F[s] -> 0, 
            y0 -> 1, yS0 -> 0}) // Simplify

ah = 1; bh = 1; y0h = 1; yS0h = 0;
((((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                            y[0] == y0, y'[0] == yS0}, y, t] // 
                        Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> 
                        k}) //.  
                (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 
                  1/(2 k)) /. {a -> ah, b -> bh, 
              F[s] -> 0, y0 -> y0h, yS0 -> yS0h}) /. 
        k -> Sqrt[ah^2 - 4  bh] // Evaluate

d

InverseLaplaceTransform[
      Evaluate[(((a y0 + s y0 + yS0 + F[s])/(b + a s + (s^2))) /. {a -> 1, b -> 1, F[s] -> 0, y0 -> 0, yS0 -> 1})], s, 
      t]

((DSolve[{y''[t] + 1  y'[t] + 1 y[t] == 0, 
                y[0] == 0, y'[0] == 1}, y, t] // 
            Flatten)) // Simplify

((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                  y[0] == y0, y'[0] == yS0}, y, t] // 
              Flatten) /. {a -> 1, b -> 1, F[s] -> 0, 
            y0 -> 0, yS0 -> 1}) // Simplify

ah = 1; bh = 1; y0h = 0; yS0h = 1;
((((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                            y[0] == y0, y'[0] == yS0}, y, t] // 
                        Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> 
                        k}) //.  
                (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 
                  1/(2 k)) /. {a -> ah, b -> bh, 
              F[s] -> 0, y0 -> y0h, yS0 -> yS0h}) /. 
        k -> Sqrt[ah^2 - 4  bh] // Evaluate

e

InverseLaplaceTransform[
      Evaluate[(((a y0 + s y0 + yS0 + F[s])/(b + a s + (s^2))) /. {a -> 1, b -> 1, F[s] -> 0, y0 -> 1, yS0 -> 1})], s, 
      t]

((DSolve[{y''[t] + 1  y'[t] + 1 y[t] == 0, 
                y[0] == 1, y'[0] == 1}, y, t] // 
            Flatten)) // Simplify

((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                  y[0] == y0, y'[0] == yS0}, y, t] // 
              Flatten) /. {a -> 1, b -> 1, F[s] -> 0, 
            y0 -> 1, yS0 -> 1}) // Simplify

ah = 1; bh = 1; y0h = 1; yS0h = 1;
((((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                            y[0] == y0, y'[0] == yS0}, y, t] // 
                        Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> 
                        k}) //.  
                (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 
                  1/(2 k)) /. {a -> ah, b -> bh, 
              F[s] -> 0, y0 -> y0h, yS0 -> yS0h}) /. 
        k -> Sqrt[ah^2 - 4  bh] // Evaluate

f

InverseLaplaceTransform[
      Evaluate[(((a y0 + s y0 + yS0 + F[s])/(b + a s + (s^2))) /. {a -> -1, b -> 1, F[s] -> 0, y0 -> 0, 
              yS0 -> 1})], s, t]

((DSolve[{y''[t] - 1  y'[t] + 1 y[t] == 0, 
                y[0] == 0, y'[0] == 1}, y, t] // 
            Flatten)) // Simplify

((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                  y[0] == y0, y'[0] == yS0}, y, t] // 
              Flatten) /. {a -> -1, b -> 1, F[s] -> 0, 
            y0 -> 0, yS0 -> 1}) // Simplify

ah = -1; bh = 1; y0h = 0; yS0h = 1;
((((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                            y[0] == y0, y'[0] == yS0}, y, t] // 
                        Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> 
                        k}) //.  
                (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 
                  1/(2 k)) /. {a -> ah, b -> bh, 
              F[s] -> 0, y0 -> y0h, yS0 -> yS0h}) /. 
        k -> Sqrt[ah^2 - 4  bh] // Evaluate

g

InverseLaplaceTransform[
      Evaluate[(((a y0 + s y0 + yS0 + F[s])/(b + a s + (s^2))) /. {a -> -2, b -> 1, F[s] -> 0, y0 -> 0, 
              yS0 -> 1})], s, t]

((DSolve[{y''[t] - 2  y'[t] + 1 y[t] == 0, 
                y[0] == 0, y'[0] == 1}, y, t] // 
            Flatten)) // Simplify

((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                  y[0] == y0, y'[0] == yS0}, y, t] // 
              Flatten) /. {a -> -2, b -> 1, F[s] -> 0, 
            y0 -> 0, yS0 -> 1}) // Simplify

ah = -2; bh = 1; y0h = 0; yS0h = 1;
((((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                            y[0] == y0, y'[0] == yS0}, y, t] // 
                        Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> 
                        k}) //.  
                (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 
                  1/(2 k)) /. {a -> ah, b -> bh, 
              F[s] -> 0, y0 -> y0h, yS0 -> yS0h}) /. 
        k -> Sqrt[ah^2 - 4  bh] // Evaluate

h

InverseLaplaceTransform[
      Evaluate[(((a y0 + s y0 + yS0 + F[s])/(b + a s + (s^2))) /. {a -> -3, b -> 1, F[s] -> 0, y0 -> 0, 
              yS0 -> 1})], s, t]

(((E^(-(1/2) (-3 + (Sqrt[5])) t)) (-1 + (E^((Sqrt[5]) t))))/(Sqrt[5])) // Expand

((DSolve[{y''[t] - 3  y'[t] + 1 y[t] == 0, 
                y[0] == 0, y'[0] == 1}, y, t] // 
            Flatten)) // Simplify

((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                  y[0] == y0, y'[0] == yS0}, y, t] // 
              Flatten) /. {a -> -3, b -> 1, F[s] -> 0, 
            y0 -> 0, yS0 -> 1}) // Simplify

ah = -3; bh = 1; y0h = 0; yS0h = 1;
((((DSolve[{y''[t] + a  y'[t] + b y[t] == 0, 
                            y[0] == y0, y'[0] == yS0}, y, t] // 
                        Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> 
                        k}) //.  
                (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 
                  1/(2 k)) /. {a -> ah, b -> bh, 
              F[s] -> 0, y0 -> y0h, yS0 -> yS0h}) /. 
        k -> Sqrt[ah^2 - 4  bh] // Evaluate

i

Remove["Global`*"]

InverseLaplaceTransform[
      Evaluate[(((a y0 + s y0 + yS0 + F[s])/(b + a s + (s^2))) /. {a -> 0, b -> -1, F[s] -> 1, y0 -> 0, 
              yS0 -> 0})], s, t]

(1/2) (E^(-t)) (-1 + (E^(2 t))) // Expand

res[
        t_] := (((E^t) - (E^(-t)))/2)

Plot[{res[t], Sinh[t]}, {t, -2, 2}];

(res[5] // N) == (Sinh[5] // N)

((DSolve[{y''[t] + 0 y'[t] - 1 y[t] == 
                  DiracDelta[t], y[0] == 0, y'[0] == 0}, y, 
              t] // Flatten)) // Simplify

((DSolve[{y''[t] + 0 y'[t] - 1 y[t] == 
                  DiracDelta[0], y[0] == 0, y'[0] == 0}, y, 
              t] // Flatten)) // Simplify

Problem in der Grenzsituation mit der Diracfunktion und den Anfangsbedingungen!

((DSolve[{y''[t] + a  y'[t] + b y[t] == f[t], 
                  y[0] == y0, y'[0] == yS0}, y, t] // 
              Flatten) /. {a -> 0, b -> -1, 
            f[t] -> DiracDelta[t], y0 -> 0, yS0 -> 0}) // 
      Simplify

ah = 0; bh = -1; y0h = 0; yS0h = 0;
((((DSolve[{y''[t] + a  y'[t] + b y[t] == 
                              f[t], y[0] == y0, y'[0] == 
                              yS0}, y, t] // 
                        Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> 
                        k}) //.  
                (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 
                  1/(2 k)) /. {a -> ah, b -> bh, 
              f[t] -> DiracDelta[t], y0 -> y0h, 
              yS0 -> yS0h}) /. k -> Sqrt[ah^2 - 4  bh] // 
      Evaluate

j

InverseLaplaceTransform[
      Evaluate[(((a y0 + s y0 + yS0 + F[s])/(b + a s + (s^2))) /. {a -> 0, b -> 1, F[s] -> 1, y0 -> 0, yS0 -> 0})], s, 
      t]

((DSolve[{y''[t] + 0  y'[t] + y[t] == 
                  DiracDelta[0], y[0] == 0, y'[0] == 0}, y, 
              t] // Flatten)) // Simplify

((DSolve[{y''[t] + 0  y'[t] + y[t] == 
                  DiracDelta[t], y[0] == 0, y'[0] == 0}, y, 
              t] // Flatten)) // Simplify

Problem in der Grenzsituation mit der Diracfunktion und den Anfangsbedingungen!

Plot[-Sin[t] + Sin[t] UnitStep[t], {t, -2, 2}];

((DSolve[{y''[t] + a  y'[t] + b y[t] == f[t], 
                  y[0] == y0, y'[0] == yS0}, y, t] // 
              Flatten) /. {a -> 0, b -> 1, 
            f[t] -> DiracDelta[t], y0 -> 0, yS0 -> 0}) // 
      Simplify

ah = 0; bh = 1; y0h = 0; yS0h = 0;
((((DSolve[{y''[t] + a  y'[t] + b y[t] == 
                              f[t], y[0] == y0, y'[0] == 
                              yS0}, y, t] // 
                        Flatten) //.  {(Sqrt[(a^2) - 4 b]) -> 
                        k}) //.  
                (1/(2 (Sqrt[ (a^2) - 4 b]))) -> 
                  1/(2 k)) /. {a -> ah, b -> bh, 
              f[t] -> DiracDelta[t], y0 -> y0h, 
              yS0 -> yS0h}) /. k -> Sqrt[ah^2 - 4  bh] // 
      Evaluate

3

VK = 4/3 10^3 Pi

VK // N

VZ = 
      NIntegrate[
        1, {x, 1, 5}, {y, -Sqrt[2^2 - (x - 3)^2], 
          Sqrt[2^2 - (x - 3)^2]}, {z, -Sqrt[10^2 - x^2 - y^2], 
          Sqrt[10^2 - x^2 - y^2]}]

V = VK - VZ

4

a

InverseLaplaceTransform[(1/((s^3) - 3  (s^2) + 3  s - 1)), s, t] // 
      Expand

DSolve[{y'''[t] - 3  y''[t] + 3  y'[t] - y[t] == 
          DiracDelta[0], 
        y[0] == 0, y'[0] == 0, y''[0] == 0}, y, 
      t]

b

LaplaceTransform[E^(-t), t, s]

InverseLaplaceTransform[(1/(1 + s)) (1/((s^3) - 3  (s^2) + 3  s - 1)), 
        s, t] // Expand

DSolve[{y'''[t] - 3  y''[t] + 3  y'[t] - y[t] == 
          E^(-t), 
        y[0] == 0, y'[0] == 0, y''[0] == 0}, y, 
      t]

c

(LaplaceTransform[y'''[t] - 3  y''[t] + 3  y'[t] - y[t], 
            t, s] /. LaplaceTransform[y[t], t, s] -> Y[s]) // 
      Expand

InverseLaplaceTransform[((1/(1 + s)) + s^2 - 3  s + 3) ( 
            (1/((s^3) - 3  (s^2) + 3  s - 1))), s, t] // Expand

DSolve[{y'''[t] - 3  y''[t] + 3  y'[t] - y[t] == 
            E^(-t), 
          y[0] == 1, y'[0] == 0, y''[0] == 0}, y, 
        t] // ExpandAll

(1/8) (E^(-t)) (-1 + 
            9 (E^(2 t)) - 10 (E^(2 t)) t + 
            6 (E^(2 t)) (t^2)) // ExpandAll

5

<< Statistics`DescriptiveStatistics`

<< Statistics`StatisticsPlots`

<< Statistics`DataManipulation`

M2 = 
      Reverse[{3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 
          2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 
          3, 9, 9, 3, 7, 5, 1}]

M1 = 
      Reverse[{0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 
          0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6, 2, 8, 0, 3, 4, 8, 2, 5, 3, 4, 
          2, 1, 1, 7, 0, 6, 8}]

Mm1 = {3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 
        2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 
        9, 9, 3, 7, 5, 1}

Mm2 = {0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 
        0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6, 2, 8, 0, 3, 4, 8, 2, 5, 3, 4, 2, 
        1, 1, 7, 0, 6, 8}

a

Frequencies[M1]

Frequencies[M2]

b

ParetoPlot[M1];

ParetoPlot[M2];

Histogram[M1];

Histogram[M2];

c

Length[M1]

LocationReport[M1] // N

DispersionReport[M1] // N

Length[M2]

LocationReport[M2] // N

DispersionReport[M2] // N

d

BoxWhiskerPlot[M1];

BoxWhiskerPlot[M2];

BoxWhiskerPlot[Transpose[{M1, M2}]];

e

Es handelt sich um je 50 Stellen der Dezimalpruchentwicklung von π.

N[Pi, 150]  - 314159/100000

N[Pi, 150]

2653589793238462643383279502884197169399375105820974944592307816406286208998628034825342

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174503778 10^30  // N

N[Pi, 50]

N[Pi, 100]

6

a

14!

14! // N

Bei Wiederholung:

14^14

% // N

b

50/7 // N

7 Binomial[50, 7] Binomial[50 - 7, 
        7] Binomial[50 - 2  7, 7]  Binomial[50 - 3  7, 
        7] Binomial[50 - 4  7, 7] Binomial[50 - 5  7, 
        7] Binomial[50 - 6  7, 7]

% // N

7 Product[Binomial[50 - k 7, 7], {k, 0, 6}]

% // N