Discharging DC Circuit Modeling Question

[n00bcake22]

Homework Statement

Hello all, I just have a quick question regarding the modeling of the following circuit (see attached).
I am trying to derive the equation for the voltage across the capacitor as a function of time. I come up with the correct result when the element polarities are the same (i.e. figure on the left) but do not when the polarities are opposite (right figure). I have attached a pdf file showing my work if you care to look it over. My resulting equations are:

Left figure:
v_c(t)=v_0*exp(-t/RC); CORRECT

Right figure:
v_c(t)=v_0*exp(t/RC); SHOULD BE NEG. IN EXP

I am a mech. eng. and thought that the polarity setup was irrelevant for modeling. Am I missing something? Thanks for the help!

Homework Equations

The Attempt at a Solution

 

[n00bcake22]

Hey Guys,

I found my error. I had inconsistent current directions--silly mistake. My apologies.