2.2

Clear[inverseLaplace]inverseLaplace[func_] := (FullSimplify@*InverseLaplaceTransform)[func, s, t]

Clear[functionList]

functionList = 3

s,

3

s + 2,

3

(s + 2)2,

3

s3,

1

2 s2 + 9,

3

s2 + 4 s + 8,

s + 4

s2 + 4 s + 8,

1

(s + 2)2;

({#, inverseLaplace /@ #} &[functionList]) // Transpose // Grid[#, Frame All] &

3s

33

2+s3 -2 t

3(2+s)2

3 -2 t t

3

s33 t2

21

2 9+s2

16Sin[3 t]

3

8+4 s+s23 -2 t Cos[t] Sin[t]

4+s

8+4 s+s2-2 t (Cos[2 t] + Sin[2 t])

1

(2+s)2-2 t t

2.3

Solve[LaplaceTransform[x[t] + 4 x[t] + 3 x[t] UnitStep[t], t, s] /.{x[0] 0, x[0] 0, LaplaceTransform[x[t], t, s] x}, x]

x 1

s 3 + 4 s + s2

Solve[LaplaceTransform[x[t] + 2 x[t] + x[t] UnitStep[t], t, s] /.{x[0] 1, x[0] 1, LaplaceTransform[x[t], t, s] x}, x]

x 1 + 3 s + s2

s (1 + s)2

Solve[LaplaceTransform[x[t] + 2 x[t] + x[t] UnitStep[t], t, s] /.{x[0] 0, x[0] 0, LaplaceTransform[x[t], t, s] x}, x]

x 1

s (1 + s)2