The impulse response of an initially relaxed linear system

The impulse response of an initially relaxed linear system is e^{– 2t} u(t). To produce a response of t e^–2t u (t), the input must be equal to—

Given
y (t) = e^{– 2t} u(t)
then

Y(s) =	1
	s + 2

and R (s) = 1 (for impulse input)
so,

H(s) =	Y(s)	=	1	=	1
	R(s)		S + 2/1		s + 2

Now, for
y(t) = te^{– 2t} u(t)
then

Y(s) =	1
	(s + 2)²

so,

X(s) =	Y(s)	=	1/(s + 2)²	=	1
	H(s)		1/S + 2		s + 2

or
x (t) = e^–2tu(t)