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From: hopp@nbs-amrf.UUCP (Ted Hopp)
Newsgroups: net.puzzle
Subject: g'(x)=g(x)
Message-ID: <491@nbs-amrf.UUCP>
Date: Sun, 31-Mar-85 22:29:04 EST
Article-I.D.: nbs-amrf.491
Posted: Sun Mar 31 22:29:04 1985
Date-Received: Wed, 3-Apr-85 02:19:58 EST
References: <917@utcsri.UUCP>
Organization: National Bureau of Standards
Lines: 18

>                                              ... A question one
> could ask is can you really find ALL functions, defined in any way,
> that equal their derivatives and  which are they?

It is straightforward to show that f(t)=c*e^t is the only function
satisfying g'(t)=g(t) and the boundary condition g(0)=c.  First, it is
clear that f satisfies the equations.  Let g be a function satisfying
g'=g, g(0)=c.  Define h(t)=g(t)*e^(-t).  Then h(0)=g(0)*e^0=c.  Also,
h'=g'*e^(-t)-g*e^(-t) (product rule for derivative).  But g'=g, so h'=0
identically.  Thus, h is the constant function c.  Thus, g(t)=c*e^t.

Of course, this assumes that h=constant follows from h'=0.  If one
wants to worry about "almost everywhere" kinds of stuff, this may
fail from time to time.  (Ugh - couldn't resist.)

-- 

Ted Hopp	{seismo,umcp-cs}!nbs-amrf!hopp