Put

Differentiate:

Now take FTs of both sides and apply the derivative rules to the forward

and backwards transform:

so we have satisfies the ODE:

and all one now has to do is show that the suggested form for satisfies

this equation (or you could just solve the equation - it's of variables separable type).

(there is at least one other way of doing it, but that requires Cauchy's integral theorem from complex analysis)

RonL