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],t,s]/.{LaplaceTransform[f[t],t,s]->F[s]})

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

((((DSolve[{y''[t]+ay'[t]+b y[t]==0,y[0]==y0,y'[0]==yS0},y,t]//Flatten)))//Simplify)

((((DSolve[{y''[t]+ay'[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)

b

a=1; b=1;F[s_]:=0; y0=1; yS0=0;
Y[s_]:=((a y0+s y0+yS0+F[s])/(b+a s+(s^2))); Y[s]

InverseLaplaceTransform[Y[s],s,t]

%//Expand

(%//N//Simplify) /.(E^(-0.5  t))->N[1/Sqrt[E],10]^ t

%//Expand//Simplify

c

a=1; b=1;F[s_]:=0; y0=0; yS0=1;
Y[s_]:=((a y0+s y0+yS0+F[s])/(b+a s+(s^2))); Y[s]

InverseLaplaceTransform[Y[s],s,t]

d

Remove[Y]

a=-1; b=1;f[t_]:=Sin[t]; y0=0; yS0=1;
InverseLaplaceTransform[Y[s]/.Flatten[Solve[(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],t,s],{Y[s]}]],s,t]

e

Remove[Y]

a=-2; b=1;f[t_]:=Sin[t]; y0=0; yS0=1;
InverseLaplaceTransform[Y[s]/.Flatten[Solve[(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],t,s],{Y[s]}]],s,t]

f

a=0; b=-1;f[t_]:=DiracDelta[t]; y0=0; yS0=0;Flatten[Solve[(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],t,s],{Y[s]}]]

a=0; b=-1;f[t_]:=DiracDelta[t]; y0=0; yS0=0;
InverseLaplaceTransform[Y[s]/.Flatten[Solve[(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],t,s],{Y[s]}]],s,t]

Expand[%]

3

Remove["Global`*"]

(LaplaceTransform[y'''[t]-3y''[t]+3y'[t]- y[t],t,s]/.{y[0]->y0,y'[0]->yS0,y''[0]->yS00,LaplaceTransform[y[t],t,s]->Y[s]})==(LaplaceTransform[f[t],t,s]/.{LaplaceTransform[f[t],t,s]->F[s]})

Solve[-(s^2) y0-s yS0-yS00-Y[s]+(s^3) Y[s]+3 (-y0+s Y[s])-3 (-s y0-yS0+(s^2) Y[s])==F[s],{ Y[s]}]

a

y0=0; yS0=0;  yS00=0; F[s_]:=LaplaceTransform[DiracDelta[t],t,s];   Y[s_]:=((3 y0-3 s y0+(s^2) y0-3 yS0+s yS0+yS00+F[s])/(((-1+s))^3)); Y[s]

LaplaceTransform[DiracDelta[t],t,s]

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

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

b

y0=0; yS0=0;  yS00=0; F[s_]:=LaplaceTransform[E^(-t),t,s];   Y[s_]:=((3 y0-3 s y0+(s^2) y0-3 yS0+s yS0+yS00+F[s])/(((-1+s))^3)); Y[s] //Together

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

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

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

c

y0=1; yS0=0;  yS00=0; F[s_]:=LaplaceTransform[E^(-t),t,s];   Y[s_]:=((3 y0-3 s y0+(s^2) y0-3 yS0+s yS0+yS00+F[s])/(((-1+s))^3)); Y[s] //Together

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

InverseLaplaceTransform[((4-2 (s^2)+(s^3))/((((-1+s))^3) (1+s))),s,t]//Expand

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

4

Remove["Global`*"]

VK=4/3 5^3 Pi

VK//N

VZ=NIntegrate[1,{x,0.5,2.5},{y,-Sqrt[1^2-(x-1.5)^2],Sqrt[1^2-(x-1.5)^2]},{z,-Sqrt[5^2-x^2-y^2],Sqrt[5^2-x^2-y^2]}]

V=VK-VZ

5

Remove["Global`*"]

(*  <<Statistics`DescriptiveStatistics`  *)

<<"StatisticalPlots`"

(*  <<Statistics`DataManipulation`  *)

N[Pi,50]

N[Pi,50]

N[Pi,100]

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}]

a

Reverse[Sort[Tally[M1]],2];

Tally[M1]//Sort

ListPlot[Tally[M1],Filling->Axis,PlotStyle->{PointSize[Large]}]

Reverse[Sort[Tally[M2]],2];

Tally[M2]//Sort

ListPlot[Tally[M2],Filling->Axis,PlotStyle->{PointSize[Large]}]

b

Needs["StatisticalPlots`"]

ParetoPlot[M1]

ParetoPlot[M2]

Histogram[M1]

Histogram[M2]

c

Length[M1]

report[x_]:={"Mean -> " Mean[x],"StandardDeviation -> "StandardDeviation[x]}

(* LocationReport[M1]//N  *)

report[M1]//N

(* DispersionReport[M1]//N  *)

Length[M2]

(* LocationReport[M2]//N  *)

(* DispersionReport[M2]//N  *)

report[M2]//N

d

BoxWhiskerChart[M1]

BoxWhiskerChart[M2]

BoxWhiskerChart[{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

Remove["Global`*"]

a

10!

10!//N

Bei Wiederholung:

10^10

%//N

b

49/7//N

Binomial[49,7] Binomial[49-7,7] Binomial[49-2  7,7]  Binomial[49-3  7,7] Binomial[49-4  7,7] Binomial[49-5  7,7] Binomial[49-6  7,7]

%//N

7Product[Binomial[49-k 7,7],{k,0,6}]

%//N