Lösungen

1. Dirachsche Deltafunktion und Einheitssprung

a

(∫_(-3 π)^(3 π))DiracDelta[
          x] \[DifferentialD]x

b

(∫_(-3 π)^(3 π))DiracDelta[
          x] E^(x + 1) \[DifferentialD]x

c

(∫_(-3 π)^(3 π))DiracDelta[
            x - (π/2)] Sin[x] \[DifferentialD]x

d

u[x_]=UnitStep[(x+1)(x-1)(x-2)];

Plot[u[x],{x,-2,3},PlotRange->{-1,3},AxesOrigin->{0,0},PlotStyle->{Thickness[0.02]}];

e

(∫_(-3)^5) u[x] \[DifferentialD]x

u[x] // FunctionExpand

f

Plot3D[UnitStep[1 - (x^2) - (y^2)], {x, -2, 2}, {y, -2, 2}, 
        PlotPoints -> 50];

2. D'Gl. 2. Ordnung

a

Remove["Global`*"]

links = 
      LaplaceTransform[y''[t] + y'[t] + y[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 1, y'[0] -> 1}

rechts = LaplaceTransform[DiracDelta[t] , t, s]

solv = Solve[links == rechts, {Y[s]}]  //  Flatten

U[s] := Y[s] /.  solv;   U[s]

U[s] // Apart

u0[t_] := InverseLaplaceTransform[U[s], s, t] // Simplify;  
    u0[t]

Plot[Evaluate[{u0[t]}], {t, 0, 6  Pi}];

b

Remove["Global`*"]

links = 
      LaplaceTransform[y''[t] + y'[t] + y[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 1, y'[0] -> 1}

LaplaceTransform[DiracDelta[t-1],t,s]

rechts = 
      LaplaceTransform[
        DiracDelta[t] + DiracDelta[t - 1] + DiracDelta[t - 2] , t, 
        s]

solv = Solve[links == rechts, {Y[s]}]  //  Flatten

U[s] := Y[s] /.  solv;   U[s]

U[s] // Apart

u0[t_] := InverseLaplaceTransform[U[s], s, t] // Simplify;  
    u0[t]

Plot[Evaluate[{u0[t]}], {t, 0, 6  Pi}];

c

Remove["Global`*"]

links = 
      LaplaceTransform[y''[t] + y'[t] + y[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 1, y'[0] -> 1}

rechts=LaplaceTransform[UnitStep[t] ,t,s]

solv = Solve[links == rechts, {Y[s]}]  //  Flatten

U[s] := Y[s] /.  solv;   U[s]

U[s] // Apart

u0[t_] := InverseLaplaceTransform[U[s], s, t] // Simplify;  
    u0[t]

Plot[Evaluate[{u0[t]}], {t, 0, 6  Pi}];

d

Remove["Global`*"]

links = 
      LaplaceTransform[y''[t] + y'[t] + y[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 1, y'[0] -> 1}

Plot[UnitStep[t]+UnitStep[t-1]+UnitStep[t-2],{t,-1,4}];

rechts=LaplaceTransform[UnitStep[t]+UnitStep[t-1]+UnitStep[t-2] ,t,s]

solv = Solve[links == rechts, {Y[s]}]  //  Flatten

U[s] := Y[s] /.  solv;   U[s]

U[s] // Apart

u0[t_] := InverseLaplaceTransform[U[s], s, t] // Simplify;  
    u0[t]

Plot[Evaluate[{u0[t]}],{t,0,2Pi}];

Plot[Evaluate[{u0[t]}], {t, 0, 6  Pi}];