Connecting a mathematical solution to a differential equation with it's physical solution 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.
