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] ?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Relationship between two solutions

**Physics Forums | Science Articles, Homework Help, Discussion**