From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: (Relatively) simple differential equation question
Date: 18 Nov 1994 18:06:50 GMT

In article <CzH46I.2sn@ul.ie>, John Kelly <kellyjn@ul.ie> wrote:
>If I have a function V(x,y) such that
[dV/dx given]
>how may I obtain a function W(x,y) such that dW     dV
>                                             __ = k __ 
>                                             dx     dx
>where k is a constant AND dW     dV
>                          __  =  __
>                          dy     dy .

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Can't do this for k=1 for (virtually) any  V: For any function  f  (with
continuous 2nd order derivatives) one has that the mixed partial derivatives
f_{xy} and f_{yx} must be equal. You're asking to have
W_{xy} = (W_x)_y = k.(V_x)_y = kV_{xy} = kV_{yx} = k(V_y)_x=k(W_y)_x=kW_{yx}
but then W_{xy}=W_{yx} too would mean (k-1)W_{yx}=0. Unless k=1, this
can only happen if W_{yx}=0, i.e., W would have to be a sum of a function of
x only and a function of y only. Same for  V, then.

Of course, if V = f(x) + g(y) then  W = kf(x) + g(y) + C is the solution.

I suppose it is possible to meet the given request with functions  V  and  W
whose derivatives are not continuous, but I don't know of any examples
(nor do I think you would want to see them...)

dave