Approach to solving a differential-functional equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:02:24Z http://mathoverflow.net/feeds/question/45530 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45530/approach-to-solving-a-differential-functional-equation Approach to solving a differential-functional equation Anixx 2010-11-10T06:57:58Z 2010-12-22T18:22:14Z <p>What could be an approach to solving such equations?</p> <p>$$f'(x)=C \prod_{k=0}^x f(k)$$</p> <p>$$\frac{g'(x)}{g(x)}=C+ \sum_{k=0}^{x-1} g(k)$$</p> <p>Here the product and the sum are understood as indefinite sum and product (i.e. generalized to real x), although a solution which holds only for integer x would also be appreciated.</p> <p>Are there any methods available? </p> <p>More than in a concrete solution I am interested to learn general methods of solving such equations.</p> http://mathoverflow.net/questions/45530/approach-to-solving-a-differential-functional-equation/45533#45533 Answer by Daniel Geisler for Approach to solving a differential-functional equation Daniel Geisler 2010-11-10T07:41:05Z 2010-11-10T07:48:20Z <p>I recognize the differential function equation you are discussing because it is produced by power towers of height x. </p> <p>Let $f(x)=a^x$, then $f^n(x)$ produces a power tower.</p> <p>Typically functional equations are solved by having some preexisting idea as to the form of the solution. I do recommend "Iterative Functional Equations" by Kuczma if you haven't read it.</p> http://mathoverflow.net/questions/45530/approach-to-solving-a-differential-functional-equation/45564#45564 Answer by Gerald Edgar for Approach to solving a differential-functional equation Gerald Edgar 2010-11-10T16:07:16Z 2010-11-10T16:07:16Z <p>Instead of "indefinite sum and product" make a change of the independent variable to convert to a conventional delay-differential equation. For example, in the first one, let $F(x) = \prod_{k=0}^x f(x)$ so that $f(x) = F(x)/F(x-1)$, then write everything in terms of $F$.<br> Of course there is no reason to think there is any "formula" for the solutions...</p>