I have a very basic issue that I can't seem to wrap my head around: I am trying to solve the membrane equation $$\tau \frac{dV(t)}{dt} = -V(t) + E(t)$$ With a time constant $\tau$, the voltage $V(t)$ and an input $E(t) = \alpha t^2$.
And after solving the homogenous version with a constat $E$ via separation of variables countless times, I fail at solving the homogenous part of this one without the $E(t)$.
The solution for $\tau \frac{dV(t)}{dt} = -V(t)$ (based on the solution sheet) is: $$V(t) = ce^{-t/ \tau}$$ But no matter how I twist it, I always end up with the following $$V(t) = e^{\frac{-t+c}{\tau}}$$ How do I get the $c$ out of the fraction in the exponent and as a factor of $E$?
Thanks for any help and apologies for the stupid question!