<< Calculus`LaplaceTransform`

Y  =  LaplaceTransform[y[t], t, s]

LaplaceTransform[Exp[t], t, s]

Yp  =  LaplaceTransform[y'[t], t, s]

Yans  = Solve[{Y  -  Yp  ==  0, y[0] == 1}, Y]

yinv  =  LaplaceTransform[y[t], t, s]  /.  First[Yans]

ans  =  InverseLaplaceTransform[yinv, s, t]