Let [itex] t \in \mathbb{R} [/itex] be large.(adsbygoogle = window.adsbygoogle || []).push({});

Let [itex] f [/itex] be a function over [itex] [0,t] [/itex] satisfying [itex] f(0) = 1 [/itex] and [itex] f'(x) = e^{-f(x)} [/itex] for all [itex] x[/itex].

Let [itex] g [/itex] be a function over [itex] [0,t] [/itex] satisfying [itex] g(0) = 1 [/itex] and [itex] g'(x) = (1 - g(x)/t^2)^{t^2} [/itex] for all [itex] x[/itex]. Note that [itex] g'(x) \sim e^{-g(x)}[/itex].

Without solving the two differential equations and finding out [itex] f [/itex] and [itex] g [/itex] (which can be done at least approximately), can the fact that [itex] f(0)=g(0)=1 [/itex] and [itex] g'(x) \sim e^{-g(x)} [/itex] be used to show that [itex] f(x) \sim g(x) [/itex] for all [itex] x[/itex] in [itex] [0,t][/itex] ?

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# Relationship between two solutions

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