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Differential equation bound for f(2003)

2175ExpertCalculusDifferential Equations

Harvard-MIT Mathematics Tournament

Two differentiable real functions $f(x)$ and $g(x)$ satisfy $$ \frac{f^{\prime}(x)}{g^{\prime}(x)}=e^{f(x)-g(x)} $$ for all $x$, and $f(0)=g(2003)=1$. Find the largest constant $c$ such that $f(2003)>c$ for all such functions $f, g$.
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