Lösungen Laplace-Transformationen

?*Laplace*

?Laplace*

?InverseLaplace*

1

a

transf = LaplaceTransform[(y''[t]+ ω  y[t]==Sin[t]),t,s]/.{LaplaceTransform[y[t],t,s]->Y[s]}

b

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

Y1[s_]:=Y[s] /. solv

Y1[s]

Y2[s_]:=Y1[s] /. {y[0]->1,y'[0]->1}

Y2[s]

c

InverseLaplaceTransform[Y2[s],s,t]

d

(InverseLaplaceTransform[Y2[s],s,t] /. ω-> 1/(2 π))//Simplify

Plot[%,{t,0,12 π}];

Plot[%%,{t,0,26 π}];

d

Ueberlagerung von periodischen Funktionen.

2

a

E^(-s Pi/2) LaplaceTransform[Cos[t],t,s]

b

LaplaceTransform[E^(t Pi/2) Cos[t],t,s]

c

LaplaceTransform[t^3 Cos[t],t,s]

d

LaplaceTransform[Sin[2t]/t,t,s]

e

LaplaceTransform[(Cos[t-1])/t,t,s]

So weit gehen die bei uns behandelten Regeln nicht...  ==> Handarbeit!

e1

LaplaceTransform[(Cos[t]-1)/t,t,s]

f

LaplaceTransform[Evaluate[Integrate[Sin[λ] Cos[t-λ],{λ,0,t}]],t,s] // Simplify

LaplaceTransform[Sin[t],t,s]*LaplaceTransform[Cos[t],t,s] // Simplify

g

f[t_]:=t-Floor[t]; Plot[f[t],{t,-3,3},AspectRatio->Automatic];

LaplaceTransform[f[t],t,s]

ρ[s_] := E^(-s*1);
1/(1 - ρ[s]) Integrate[
        E^(-s τ) * τ, {τ, 0, 1}]