Example 6.3.4

Plot[Sin[t-3]HeavisideTheta[t-3],{t,0,10},PlotStyle->Thick]

Compare the following two animations to see how the shiting of sin t affects the sketch.

Manipulate[Plot[Sin[t-c]HeavisideTheta[t-c],{t,0,10},PlotRange->{{0,10},{-1,1}},PlotStyle->Thick],{c,0,10}]

Manipulate[Plot[Sin[t]HeavisideTheta[t-c],{t,0,10},PlotRange->{{0,10},{-1,1}},PlotStyle->Thick],{c,0,10}]

Example 6.3.7

f[t_]=Piecewise[{{0,t<2},{(((t-2))^2),t≥2}}]

LaplaceTransform[f[t],t,s]

Plot[f[t],{t,0,4},PlotStyle->Thick]

Plot[HeavisideTheta[t-2](((t-2))^2),{t,0,4},PlotStyle->Thick]

Integrate[Exp[-s*t](((t-2))^2),{t,2,∞}]

Example 6.3.18

F[s_]=(1/s)(Exp[-s]+Exp[-2s]-Exp[-3s]-Exp[-4s])

InverseLaplaceTransform[F[s],s,t]

Plot[%,{t,0,10},PlotStyle->Thick]