How to determine a system is stable without knowing the input?
Seemingly a homework.

Unstable system starts to generate increasing output if there exists some excitation other than a sometimes possible stabilizing one that is generated by external controller. Thus c is in the very beginning false. You must solve "a or b".


Laplace transform the left and right side
ignore the initial value terms
replace the right side with 1 (it's all the same, what is the input)
solve Y(s), you get a rational function of s
solve the poles of Y(s) (= values of s that make the Y(s) indefinite due the division by zero)
system is stable if all poles are in the left half of the complex s-plane


Imaginary poles are a limit case. The output continues to swing if some excitation has happened, but does not expand in amplitude. You must check, is it considered stability or unstability in your definitions.