These are not at all stupid questions!...

First answer: the general solution of the DE is...

(1)

... and that means that is a second order linear DE with constant coefficients and its 'characteristic equation' has solution with molteplicity two, i.e. is...

(2)

The corresponding DE is...

(3)

Second answer: the interval in which a linear second order DE has one and only one solution for any 'initial condition' is the inteval in which the 'coefficient' of the term doesn't vanish, i.e. . In a) is , in b) is , so that...

Kind regards