hopp@nbs-amrf.UUCP (Ted Hopp) (04/01/85)
> ... 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