Step 1

To determine:

The existence of Laplace transforms of the functions:

a) \(f(t)=t^2\sin(\omega t)\)

b) \(f(t)=e^{t^2}\sin(\omega t)\)

Step 2

Definition of existence of Laplace transforms :

Let f be a piece – wise continuous function in \([0,\infty)\) and is of exponential order. The Laplace transformation F (s) of f exists for some \(s > c\), where c is a real number depends on f.

Step 3

a)Since, the function \(f(t)=t^2 \sin(\omega t)\) can be expressed as an exponential order. Hence the Laplace transform exists for the function \(f(t)=t^2 \sin(\omega t)\)

Step 4

b)The \(f(t)=e^{t^2}\sin(\omega t)\) cannot be expressed as an exponential order, since it grows too fast than the exponential function and bounded by \(e^{-t}\) and hence, the integral doesn’t converge

it would at some point exceed the \(e^{-st}\) term.

This is the reason for the Laplace transformation of this function is undefined.