Are Anti-Derivatives Unique?

Two differentiable real functions $f,g$ satisfy $f'(x)/g'(x) = e^{f(x) - g(x)}$ for all reals $x$, and $f(0) = g(2003) = 1$. There is a largest constant $c$ such that $f(2003) > c$ for all such functions $f,g$. Find $e^{1 - C}$.

Source: HMMT February 2003