I have seen this question in a neuroscience course:
It is given after the lecture with these and these slides.
I have no background in physics. However, I do know how to solve a differential equation of this form. It is a linear first order differential equation.
So, here is my solutions for the cases of $I=0$ and $I=const=I_0$:
However, I can't connect the two solutions (mine and the one given in the course).
In other words, how come all 3 answers are correct?
where does the $T(t-s)$ come from?
where does the 9.5 come from?
How can I get from the question that I have to integrate from $0$ to $t$ or from $0$ to $infty$?.
$9.5-u_{rest}$ can be accounted for by your constant $C_0$. Since you already can see why the third option is correct (actually directly from the question by using the integrating factor), you can perform a substitution to get the first option, and then add the homogenous solution to get the second option.