[MATH]\int \dfrac{ds}{S} = \int -a \ dt \implies\\
ln(S) + c_1 = - at + c_2 \implies ln(S) = - at + c,\\
\text {where } c = c_2 - c_1.\ \checkmark[/MATH]Now you have a set of functions, differing by the value of c. Can you determine c? Well, what else do you know?
[MATH]S(0) = S_0.[/MATH] You are given that in the statement of the problem, correct?
[MATH]\text {Therefore, it must be true that } ln(S_0) = -a(0) + c \implies\\
c= ln(S_0).[/MATH]Is that not obvious?
[MATH]ln(S) = - at + c = -at + ln(S_0) \implies\\
- at = \ln(S) - ln(S_0) = ln \left ( \dfrac{S}{S_0} \right ) \implies\\
\dfrac{S}{S_0} = e^{-at} \implies S = S_0 * e^{-at}.[/MATH]There must be something in your text that talks about separable differential equations. But you just solved one even if you did not understand the text..
Now where is your problem with part b?